I am in no way an expert on auto claims but I recently had to file one myself. I called my insurer immediately after the fender bender and asked if I needed to notify the police. They said that it was my option as to whether or not I wanted a report filed. When there is no police report you can still file a claim but the insurance company is now responsible for determining who is at fault. They generally do this by examining the damage and taking statements from both drivers and any passengers or noted witnesses.
I hope that this helps and if you need any information on retirement, college savings, investments, or life insurance feel free to ask since those are my main area of expertise. Best of luck!