Are you supposed to report a car accident where there was no damage?

It isn't illegal not to report it. There are times when people with just a scratch on the bumper will report it to the other person's insurance company and claim massive injuries from getting tapped in traffic. If you don't call the police, it is a good idea to at least take photos of the damage after an accident. The guy could go home, back into something to cause more damage, then call your insurance company to claim damages. More than likely this will not happen, just saying it does every now & then.