I would respectfully disagree with Hawkmoon. My wife has been hit three times in the last 7 years. The first time we did as Hawkmoon has suggested, and called the person at fault's insurance and played their stupid games. The gave us the run around, gave us crap about the shop we picked, and all in all didn't give a damn how we felt about the process. Which I understand, as we aren't their customer, but it's still annoying. Then one of my friends pointed out that we pay insurance premiums for a reason.
The next two accidents, we called our Allstate agent, and told them to fix it. They towed the car, had it fixed, went after the at fault people, and got my deductible back. (one of the deductibles was refunded before I actually paid it, the other one a few weeks later) They have folks on staff whose job it is to run down owners, owner's insurance companies, file the correct forms in triplicate and whatever else is needed to get my (or your) money back. Since you pay those folks, why would you not use them? AN added benefit is that since you are your insurance companies customer, they actually care that you are happy at the end of it.
To answer your question, since the owner is a known entity, your insurance will almost certainly attempt to recoup the repairs from either them, or their insurance. Your auto insurance company is not in the business of eating the cost of repairs when the responsible party is known, and likely insured.