Auto Insurance Claims: Auto Vehicle Accident, car insurance companies, other vehicles

Question
This afternoon my brother was involved in a vehicle accident.  From his scenario, he was at a park and backing his car out of a parking place to repark when a vehicle appeared out of no where and the back end of my brothers vehicle and the rear passenger of the other drivers vehicle collided.  After speaking with the other driver, he admitted to not seeing my brothers vehicle.  Had the other vehicle tried to stop, swerve, honk, or otherwise get out of the way, or alert my brother in some way, they could have had a less serious accident, however, since the other driver had not seen my brother backing up, he continued driving even after they had hit eachother, thus scraping up the whole back bumper of my brothers car as well as the whole back passenger door and bumper of his own car.  The police were called to make a report, but no tickets were issued.  The reason given for that was because it was in a park and on private property, and tickets could not be written there.  My brother does not intend on fixing the scratches on his vehicle, but there is substantial damage on the other vehicle.  My question is, how do car insurance companies handle a claim where there is no tickets written and from what could be carelessness on both drivers part?  How do you determine whose fault the accident would be and who would have to pay for the damage?

Answer
Tickets have to do with breaking traffic laws.  One can still be labile without breaking a law though.  
The scenario that you've given does place the majority of the due care on the part of your brother, not the other vehicle.  Whoever is backing or otherwise leaving a parking spot must yield right of way to other vehicles.  If the other vehicle had a clear chance to avoid the accident, then in some states, they may be held partially liable.  Your insurance company should be notified so they can properly defend you and access the proper liability.
Good luck!