Can someone else make a claim on my car insurance?

Google "subrogation."

Having had a number of accidents over many years of driving (don't ask), let me say that this scenario is why you have auto insurance. While it may at times be useful to avoid filing a claim for very minor damage (to avoid having your rates go up if you've had a number of accidents), this person is being impossible. Say nothing and sign nothing. Report the fenderbender ASAP to your insurer and tell this person to contact their insurance company and/or your insurance company and to not call you again. The insurance adjusters will quickly hash this out, cut any damage estimate down to size and be done with it. Insurance estimates (at least in my experience in New York) are always inflated and just a starting point for negotiations between an insurance company and the body shop that will be fixing a car. It's a game. Also, note that insurance companies now "record" your accident description when you call. This avoids filling out a bunch of forms and pins everyone down on their story. Think about what you want to say before you call. Relax and don't worry. This sounds like a trivial matter. :)

Yes, of course the other driver can contact your insurance company. Usually his insurance agent will do this for him. If you say it was not your fault and there are no witnesses or police report on the accident, your insurance company will probably refuse to pay anything to the other driver. However, the fact that you made an offer to settle suggests that you did admit fault. In any case, don't go changing your story now. If you think it was your fault, you should take responsibility. If you think it was not your fault and you stated that at the time, your insurance company will take on the job of trying to prove or disprove that. Your insurance agent will not be happy with you if the other driver calls them before you report the accident. It would be best to contact your agent now and explain the situation. http://info.insure.com/auto/collision/afteraccident.html is some helpful advice on how to deal with accidents in the future.

Also consider that the other person has zero incentive to find the cheapest estimate.

Also consider that the amount of damage in a low speed crash can easily top $1000 up to $2000, especially if fixed at the dealer. Personally, I think it's a scam perpetrated by the auto manufacturers, but http://www.iihs.org/ (Insurance Institute of Highway Safety) has been documenting this for years. You can find official results on their site, but here is a http://www.theautochannel.com/search/search.html?method=and&format=TachNet&restrict=&exclude=&config=htdig&words=low+speed+crash+results.

Thanks for all these responses. Just to clear a couple of things up: the police were not called; there was no police report filed; there was no damage to my car; and the damage to the other car was basically some scraped paint. Although I offered to settle the matter to avoid any further inconvenience, I am not persuaded that it was my fault. I guess I am skeptical of the claim that I have an obligation to report something like this to my auto insurance company. My understanding is that I'm free to not report something like this; the other person can sue me and get a judgment, if a court determines I was at fault, and I can be compelled to pay a judgment---but otherwise, the other person can't do anything as long as I deny that I am at fault. This happened a week ago. Yesterday, the other person threatened to go ahead and make a police report now, and then file a claim with my insurance company. But how would the police make any determination of who was at fault, a week after the fact?

I would report it. I was on the other side of a very similar situation a couple of years ago, and did make a claim on the other guy's insurance directly (not through my insurer.) The representative I talked to was somewhat taken aback that this was the first they had heard of the accident and implied that Mr. SUV's rates were going to rise because he was at fault and hadn't let them know. It's not just your insurance carrier you should be worried about either. I don't know what the law is in other states, but here in CA you are required to report accidents that result in injury or property damage greater than $750 to the DMV within 10 days, whether they occurred on public or private property and regardless of fault. Failure to report can result in a license suspension. (http://www.dmv.ca.gov/dl/dl_info.htm#2509) As to the cost, body shops give tremendous discounts to insurance companies (or, if you like, tremendously inflate prices for the average consumer.) I got walk-in estimates ranging from $2200 to $2800 for a wrinkled fender and hood on an 11-year-old Civic -- his insurance company got the job done for less than $1500 at a shop that had estimated the job for me at $2500.

I'm not sure where you are, but my understanding is that here, in Ontario, they'd make a claim with their insurance company, who would then contact your insurer, and come to some agreement with your insurer as to who was at fault for the accident. Then either your insurer or theirs would pay the claim.

...there was no damage to my car; and the damage to the other car was basically some scraped paint.And if you're lucky, like me, the other driver will claim you totaled their car, you will protest, your insurance company will fail to investigate, you will indicate fraud is taking place, your insurer will ignore you, the other driver will have his car "totaled" and get $2,000 for a $20 scratch on a car that can't be worth more than $750. People are strange -- insurance companies more-so.