This phase III trial studies how well chemotherapy and radiation therapy (chemoradiation) with or without atezolizumab works in treating patients with limited stage small cell lung cancer. Drugs used in chemotherapy, such as etoposide, cisplatin, and carboplatin, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Radiation therapy uses high energy x-rays to kill tumor cells and shrink tumors. 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. Giving chemoradiation with or without atezolizumab may work better in treating patients with limited stage small cell lung cancer.