Liability in a car accident revolves around who is deemed at-fault. As the owner of a stolen vehicle is the victim of a crime they cannot be held responsible if said vehicle gets into an accident. Therefore, the liable insurance company in this instance would be yours. Even though the car thief is at-fault insurance does not cover illegal acts, so they will have no way to compensate you. Any damage done to your vehicle will need to be covered by your insurer. Luckily, this will not reflect badly on you because you are also a victim.
Typically, in a accident like this you will file a claim with your insurance for the damages. The company will issue you a payout so you can make repairs, and then the insurance company will seek satisfaction from the car thief. Neither you nor the actual owner of the car will have to bother with this as it will be solely between the insurer and the thief. There is no risk in your claim not being accepted as you are the victim of a crime which is fully covered in comprehensive coverage. Additionally, your rate will not be affected either.
There are small details that cab affect his. If the thief is related to the car owner that changes things a bit. A family member behind the wheel does assign partial liability. Also, if you are held at fault in any way you may have to pay your excess. At-fault is dictated by the state you happen to be in and not the insurance company. Some states do not assign total fault to just one driver, so there could be a little bit of a slippery slope. If you are not in such a state then you do not have to worry about it.