IF the damage is minor and not too expensive, and IF the two of you can come to an amicable agreement over fault and payment, then I'd try to arrange something privately and leave the insurance company totally out of it. It all really depends on the personality of the guy and the laws where you are. Usually there is some minimum amount of damage before the police must get involved. Same often goes for insurance (eg. deductible sometimes higher than damage.) There's nothing that says you must get insurance involved for every little thing.

By not claiming it, the person at fault can keep their record "clean" and avoid possible higher insurance payments later. Of course, some folks don't care and will fight tooth and nail to try and get out of taking responsibility. In that case, you'll have to get the insurance company involved.

Many years ago a neighbour of mine backed into my parked car, causing a dent in the driver's side front door. She left a note, apologised, and offered to pay for it out of her pocket. My mechanic gave me a quote of $200 CAD to fix it, and the neighbour was happy. No fuss, no bother. Got fixed, neighbour paid, no insurance involvement. But that was an ideal situation.

Good Luck!