Can a police officer give you a ticket at your house?

Sure. Police officers rarely actually see the accident happen. They draw a conclusion from what the two drivers and any other witnesses say and the physical evidence and decide whether to give anyone a ticket. That the officer gives one driver a ticket after that driver has left the accident scene is irrelevant. If it mere not, a negligent driver could just flee the scene before the police arrived in order to avoid the scene [but that then creates the crime of fleeing the scene of an accident]. Your son should be the one asking this question, as he has the knowledge of what occurred and has to bear the responsibility of what he did.