Generally, property damage claims are resolved fairly quickly and easily. But every now and then, you’ll get an issue where the you aren’t satisfied with the way the other driver’s insurance is handling your property damage claim. For example, you may not trust their repair estimate or you may not agree with their decision to total (or not total) your vehicle.
If you have property damage coverage on your car, the best way to deal with this is to go through your insurance company. At a minimum, you can have your insurance company send an adjuster out, and you can see if your company’s adjuster agrees with the other driver’s insurance company.
If they agree, then maybe you feel confident enough to proceed using the other driver’s insurance. But if they don’t agree (or you don’t have confidence in the other company), then you can make a claim on your own policy. Yes, you’ll be out your deductible. But when your insurance company tries to recover its payment from the other insurance company, your insurance company will also recover your deductible for you.