If you're in a no fault state, you have no recourse but to call your insurance. If you're not in a no fault state, and are in a hurry you can call your insurance, file a claim, pay your deductible, and then let them subrogate the claim and see about getting you your deductible back. However, if you're trying to not muddle up your insurance claims, and you have someone who is at fault, and their insurance, then absolutely you should call their insurance company, on their claim and get it fixed. Thats why states at a minimum require individuals have liability insurance to cover the damages of others.