We recently had a incident where one of our cars was on loan, and someone else reversed into the door of it. Names and addresses where exchanged and the other party admitted liability. we got a cash price of getting the door repaired of £400, which they decided was too much and wanted to put it through their insurance.
So we notified our insurers but put the claim in the hands of an accident management company. They chased the claim through and got it all sorted.
But always tell your insurer and advise them that you are not proceeding with a claim through them, it has been known that the other party will put a counter claim in and claim for personal injury as well!!!