Whose insurance is responsible for damages on a loaner car from a car dealership?

Each state has its own insurance laws and regulations.  Here in Texas, your automobile insurance covers you while you are operating any vehicle, regardless of ownership.  That is why, when you rent a car, you must either provide proof of your own insurance, or allow the rental company to place coverage on you while are operating their car.  That is also why, when you want to test drive a car at a dealership, the salesperson asks to copy your driver's license and insurance card.  The same applies to loaners.  Yes, the dealer has a blanket garagekeeper's liability policy that covers their vehicles, but it only kicks in if your insurance is either non-existent, out of force, or insufficient to cover any damage you cause while operating their vehicle.  And, in that case, the dealer and/or the dealer's insurance company can sue you for any damages that your insurance does not cover.