Evaluating the Addition of the Immunotherapy Drug Atezolizumab to Standard Chemotherapy Treatment for Advanced or Metastatic Neuroendocrine Carcinomas That Originate Outside the Lung

Inclusion Criteria: * Participants must have histologically-confirmed (local site pathological confirmation sufficient) extrapulmonary poorly differentiated, neuroendocrine carcinoma (NEC) * Participants must have disease that is unresectable or metastatic and not eligible for definitive therapy as deemed per the treating investigator * Participants must have radiologically evaluable disease, measurable or non-measurable, per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria. All measurable and nonmeasurable lesions must be assessed by CT scan with IV contrast of the chest/abdomen/and pelvis (or CT chest without contrast and MRI abdomen/pelvis with gadolinium contrast, if contraindication to CT iodinated contrast) within 28 days prior to registration. While may be used for routine clinical evaluation, PET scans and bone scans alone are not acceptable for disease assessment while participating in this study. All known sites of disease must be assessed and documented on the Baseline Tumor Assessment Form * Participants must have brain MRI (or CT head with contrast if there is contraindication to MRI brain) if clinically indicated within 28 days prior to registration. Note: Brain imaging is not required in participants without known and/or clinical concern for brain metastases. Participants with asymptomatic central nervous system (CNS) metastases are eligible if one or more of the following apply: * Participants who have received treatment for brain metastases must have: * No evidence of radiological progression (by MRI brain or CT head with contrast if there is contraindication to MRI brain) within 28 days prior to registration * Discontinued all corticosteroids at least 14 days prior to registration * Participants with treatment-naive brain lesions must have: * No lesion measuring \> 2.0 cm in size in any axis * MRI brain or CT head with contrast (if there is contraindication to MRI brain) demonstrating no evidence for mass effect, edema, or other impending neurological compromise within 28 days prior to registration * No evidence of radiological progression (by MRI brain or CT head with contrast if there is contraindication to MRI brain) within 28 days prior to registration * No need for \> 2 mg of dexamethasone (or equivalent of \> 10 mg prednisone) per day at time of registration * Participants must not have symptomatic central nervous system (CNS) metastases * Participants must not have known or suspected leptomeningeal disease * Participants with prior history of non-metastatic (localized/locally advanced disease) extrapulmonary poorly differentiated NEC may have had prior platinum-based therapy +/- radiation +/- surgery provided that all therapy was completed \>= 6 months prior to registration * Participants must discontinue denosumab prior to study registration and plan to replace with a bisphosphonate while on the study * Participants must not have had prior treatment for advanced or metastatic NEC EXCEPT one cycle of platinum (carboplatin/cisplatin) + etoposide is allowed prior to registration. Other chemotherapy regimens are not allowed. For participants with prostate or urothelial NEC, prior chemotherapy for the non-NEC component (e.g. adenocarcinoma or urothelial) is allowed as long as such therapy was completed \>= 24 weeks prior to registration and participants have recovered from all prior toxicities to =\< grade 1. * Participants must not have had prior treatment with an anti-PD-1, anti-PD-L1, antiPD-L2, CD137 agonists, anti-CTLA-4 agent, or any other immune checkpoint inhibitors for any neuroendocrine neoplasm. Immune checkpoint inhibitors given for other cancer indications are allowed provided last therapy was given at least 12 months prior to study registration * Participants must not have received treatment with systemic immunostimulatory agents including, but not limited to, interferon and interleukin2 \[IL-2\] within 4 weeks or 5 half-lives of the drug (whichever is longer) prior to registration * Participants must not have had history of known severe allergy, anaphylactic, or other hypersensitivity reactions to chimeric or humanized antibodies, including to Chinese hamster ovary cell products or to any component of the atezolizumab formulation, cisplatin, carboplatin, or etoposide * Participants must not be on active systemic therapy for another cancer with the exception of hormonal therapy including androgen deprivation therapy (e.g., gonadotropin-releasing hormone \[GnRH\] agonists or antagonists), which can be continued while participants are receiving protocol therapy. Use of enzalutamide or apalutamide is permitted after completion of chemotherapy and must be held during chemotherapy for participants receiving prior to enrollment. Use of darolutamide is permitted during chemotherapy. Glucocorticoid-containing regimens, including abiraterone, are not permitted. * Participants must be \>= 18 years of age * Participants must have a Zubrod performance status of =\< 2 within 28 days prior to registration * Participants must have a complete medical history and physical exam within 28 days prior to registration * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9 /L (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Hemoglobin \>= 9.0 g/dl (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Platelet count \>= 100 x 10\^9/L (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Serum alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x institutional upper limit of normal (ULN) (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Serum total bilirubin =\< 1.5 x ULN (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Adequate renal function as defined by any 1 of the following: 1) Measured creatinine clearance (CL) \> 50 mL/min OR 2) Calculated creatinine CL \> 50 mL/min by the Cockcroft-Gault formula OR by 24-hour urine collection for determination of creatinine clearance (obtained within 14 days prior to registration. For participants who received a cycle of chemotherapy prior to registration, at least 21 days must have elapsed between day 1 of platinum + etoposide and performance of these tests) * Participants must not have uncontrolled or symptomatic hypercalcemia (\> 1.5 mmol/L ionized calcium or calcium \> 12 mg/dL or corrected serum calcium \> ULN) within 14 days prior to registration. Participants who have asymptomatic hypercalcemia are eligible provided that medical therapy to treat the hypercalcemia is planned * Participants must not have a diagnosis of immunodeficiency nor be receiving systemic steroid therapy (equivalent of \> 20 mg of hydrocortisone per day) or any other form of immunosuppressive therapy within 14 days prior to registration * Participants must not have active or history of autoimmune disease or immune deficiency, including, but not limited to myasthenia gravis, myositis, autoimmune hepatitis, systemic lupus erythematosus, rheumatoid arthritis, inflammatory bowel disease, antiphospholipid antibody syndrome, Wegener granulomatosis, Sjogren syndrome, Guillain-Barre syndrome, or multiple sclerosis with the following exceptions: * Patients with a history of autoimmune-related hypothyroidism who are on thyroid-replacement hormone are eligible for the study * Patients with controlled type 1 diabetes mellitus who are on an insulin regimen are eligible for the study * Patients with eczema, psoriasis, lichen simplex chronicus, or vitiligo with dermatologic manifestations only (e.g., patients with psoriatic arthritis are excluded) are eligible for the study provided all of following conditions are met: * Rash must cover \< 10% of body surface area * Disease is well controlled at baseline and requires only low-potency topical corticosteroids * No occurrence of acute exacerbations of the underlying condition requiring psoralen plus ultraviolet A radiation, methotrexate, retinoids, biologic agents, oral calcineurin inhibitors, or high-potency or oral corticosteroids within the previous 12 months * Participants must not have history of idiopathic pulmonary fibrosis, organizing pneumonia (e.g., bronchiolitis obliterans), drug-induced pneumonitis, idiopathic pneumonitis, or evidence of active pneumonitis on screening chest CT scan. NOTE: History of radiation pneumonitis in the radiation field (fibrosis) is permitted * Participants must not have significant cardiovascular disease, such as New York Heart Association class II or greater cardiac disease, myocardial infarction within 3 months prior to registration, unstable arrythmias, or unstable angina * Participants must not have had a major surgical procedure other than for diagnosis within 28 days prior to registration. Participant must not plan to receive a major surgical procedure during the course of protocol treatment. NOTE: Patient port placement is not considered a major surgery * Participants must not have severe infections (i.e., Common Terminology Criteria for Adverse Events \[CTCAE\] grade \>= 2) at time of registration, including but not limited to hospitalization for complications for infection, bacteremia, or severe pneumonia * Participants must not have known active tuberculosis * Participants with evidence of chronic hepatitis B virus (HBV) infection must have undetectable HBV viral load, with testing performed as clinically indicated * Participants with a history of hepatitis C virus (HCV) infection must have been treated and cured. Participants with active HCV infection who are currently on treatment must have an undetectable HCV viral load, with testing performed as clinically indicated * Participants with known human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy at time of registration and have undetectable HIV viral load within 6 months of registration * Participants must not have prior allogeneic bone marrow transplantation or solid organ transplant * Participants must not have received administration of a live, attenuated vaccine (e.g., FluMist \[registered trademark\]) within 28 days prior to initiation of study treatment, during treatment with atezolizumab, and not plan to receive for 5 months after the last dose of atezolizumab * Participants must not be pregnant due to the possibility of harm to the fetus. Individuals who are of reproductive potential must have agreed to use an effective contraceptive method (with details provided as a part of the consent process) during the treatment period and for 5 months after the final dose of atezolizumab. A person who has had menses at any time in the preceding 12 consecutive months or who has semen likely to contain sperm is considered to be of "reproductive potential." In addition to routine contraceptive methods, "effective contraception" also includes refraining from sexual activity that might result in pregnancy and surgery intended to prevent pregnancy (or with a side effect of pregnancy prevention) including hysterectomy, bilateral oophorectomy, bilateral tubal ligation/occlusion, and vasectomy with testing showing no sperm in the semen * Participants must be offered the opportunity to participate in specimen banking. With participant consent, specimens must be collected and submitted via the Southwest Oncology Group (SWOG) Specimen Tracking System * Participants must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines