You don’t have to use the other party’s insurance company to pay for your property damage. Some insurance companies are actually telling clients to do that. You might call your company and they will tell you that you have to use the company of the responsible party to fix your car.
However, if you have your own insurance company’s comprehensive and collision coverage, you should be able to turn to your own insurance company and have them fix your vehicle. In fact, this is usually the best way to go. I’m not sure why the insurance companies are doing this (passing the buck), but I’ve been hearing this story more often than not. You have every right turn to your own company to fix your own vehicle. You will incur a deductible, but your insurance company will reimburse that to you eventually in most cases. There might be some rare cases where your insurance company won't impose the deductible. Just ask them if they'd consider waiving the deductible, especially when it's pretty certain they're going to recover the money from the opposing insurance company.
After they fix your car, they will demand repayment from the other party's insurance company. If they can't resolve their claim, they might enter into arbitration - this is the process by which insurance companies resolve their property damage disputes. Suffice it to say that when they get repaid, they should reimburse your deductible to you at that time. You might have to check with them every so often about that. Believe it or not, sometimes they forget but they usually do remember to reimburse you.