This phase II trial studies how well atezolizumab works alone or in combination with etrumadenant or tocilizumab in treating men with localized prostate cancer before radical prostatectomy. 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. Androgens can cause the growth of prostate cancer cells. IL-6 is expressed by prostate cancer and within the tumor microenvironment and shown to enhance prostate cancer and disease progression. Treatment with an anti-IL-6 antibody such as tocilizumab may inhibit cancer progression. Giving atezolizumab in combination with etrumadenant or tocilizumab may work better in treating prostate cancer.