Asked by user in New York.
As I was parking my car, I was close to another car. The other driver said I scratched his car and he started to take pictures. His car was very old, but he asked me for $20 for the scratch. I said that I didn't touch his big van, and he took the money and I just left.
It's always a good idea to get a police report after an accident, no matter how small, and also to report it to your insurance company. A report to the insurer is different from a claim. Even if you're planning for paying for accident damage out-of-pocket, having a report on file with your insurer can protect you if the other party decides to file a claim.
In short, yes, if you damaged the other car, the driver can follow up in a variety of ways if he has your license plate number or other identifying information. For one thing, if you didn't exchange information, he could report the incident as a hit and run. Second, he could make a claim to his insurance company if there's damage that warrants a repair.
If his vehicle was older, as you mentioned, and perhaps not in perfect condition, it would be difficult for him to prove that a particular scratch or dent was the result of the accident. But, in general, it protects all parties to have a police report filed and make a report to the insurance company, even for a minor incident.