This phase II/III trial compares the effect of immunotherapy with atezolizumab in combination with standard chemotherapy with a platinum drug (cisplatin or carboplatin) and etoposide versus standard therapy alone for the treatment of poorly differentiated extrapulmonary (originated outside the lung) neuroendocrine cancer that may have spread from where it first started to nearby tissue, lymph nodes, or distant parts of the body (advanced) or that has spread from where it first started (primary site) to other places in the body (metastatic). The other aim of this trial is to compare using atezolizumab just at the beginning of treatment versus continuing it beyond the initial treatment. Immunotherapy with monoclonal antibodies, such as atezolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Cisplatin and carboplatin are in a class of medications known as platinum-containing compounds that work by killing, stopping or slowing the growth of cancer cells. Etoposide is in a class of medications known as podophyllotoxin derivatives. It blocks a certain enzyme needed for cell division and DNA repair, and it may kill cancer cells. Giving atezolizumab in combination with a platinum drug (cisplatin or carboplatin) and etoposide may work better in treating patients with poorly differentiated extrapulmonary neuroendocrine cancer compared to standard therapy with a platinum drug (cisplatin or carboplatin) and etoposide alone.