Liability should be pretty simple. Just like now, the owner of the car is liable. In my state this is the case even if someone else is driving; not sure this is the case elsewhere but that would seem to apply to a computer driver as well as a human one. The logic is, if you allow someone else to use your property and they cause damage with it, you are liable.

Insurance (assuming the owner has it in force) may pay, but the liability still rests with the owner.