My very good horse-sitter, and cat-sitters I've used in the past, and most of the barns where I've boarded horses over the years, have you sign a paper that says basically: I am authorizing this person to contract for emergency veterinary work done on the animal if the person deems it necessary and can't get in touch with me with a single phone call; and I agree to pay any and all vet bills that result. But that if it's NOT an emergency, the sitter will wait til she can get in touch with me before having anything done.
With my horse-sitter I also have a clause that if the horse's expected chance of long-term pasture-soundness or chance of comfortable survival for at least 1 year is less than 50%, they have my permission for euthanasia instead. (Could substitute whatever the owner's individual wishes are, and I'm not sure how this would map to dogs instead of horses)
I'm not a lawyer, and I'm not saying whether this is technically necessary/sufficient, but short of GOING TO an actual lawyer in your state, it seems like a reasonable thing to have in a contract.
JME,
Pat