In most states in the US (maybe true for ALL states) - the claim goes thru the guilty party's insurance. The other party's insurance has no role to play in it. If you have to go through YOUR insurance regardless of whose mistake it was, then what is the point in determining who was at fault?
I have had some brushes with insurance companies, covering many different scenarios - I have been on both sides of the claim - am pretty sure I know the rules now!!
The important point is - don't make mistakes, don't hit anyone.