I don't see how this is possible given there are some places where police won't even respond to fender-benders unless injuries or alcohol are involved.
I wasn't aware of that, I'm guessing you still have to call it in? I was rear ended once and it was similar to your situation - just a little ding - so I just got her insurance info. The insurance company refused to pay. Although it didn't matter since I totaled my car a couple of weeks later when someone pulled out in front of me, and then I got a new car and totaled that a week later by someone pulling out in front of me. This was 15 years ago though.