I can agree a bit with the woman.
I dealt with a 'I will pay you' accident before. It took 6 months to get $500. My frend ended up paying a little more than that even with me getting the parts cheap and replacing them...then having a body shop blend it in. It would have been double had he had them do it all or more. It's rare an accident is only $500 now.
Personally, and it's still a risk; if they want to pay out of pocket I will ask let's go get the money...snap shots with my camera phone prior. If it's after banking hours, it's a phone call to the po po then report to my insurance. Most police don't do real accident reports today though...still I want them there just for having a witness to events.
I believe it's sort of the law to report all accidents anyway.
I have read one internet story that was valid of a person agreeing to settle on the side and the one that was hit became the one that caused the accident then 'fled'. It was a nightmare for them. It was years ago though. They went through a lot of BS though and in the end the idiot that caused it all only had to pay damages to fix their car....nothing on the ordeal.