If you live in an area where insurance is optional, then it is between you and the person damaged. I see no "third party". This is very risky business - if the only damage is to property, then you are very lucky because injuries could cost you hundreds of thousands to a couple of million. If you were at fault, the other person could sue for damages. If you live where insurance is mandatory, then you will be hostage to whatever they ask for damages - or they could report you to the state.
I believe that your obligation is to "make the other party whole" and if they are only asking for repair work, you are getting off easy, unless you believe the estimate is totally fraudulent. In addition to the cost of the repairs they could ask (in court) for diminished value as the repaired vehicle will be worth less, sometimes much less, than a vehicle with no accident history.
You should see a lawyer to make sure that when you pay for the damage, that will be the end of it. Otherwise, there could be more claims against you for hidden damage, further repairs, diminished value, car rental during the repair, undisclosed injuries, pain & suffering etc.
When its over, buy insurance.
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