Who is responsible by law for paying vet bills if a dog attacks another dog?

It depends. If there was a chance the dog on the leash was dangerous and there was a possibility of an attack, it should have been put into a mouth guard, to protect other dogs/individuals. Some dogs may not be overally dangerous, but just protective of their owners. However, the dog off the leash is only allowed to be off in Parks etc which allow it to be off. This means though, that the dog must be able to be controlled, and should follow the commands of a significant other-it's owner. So if there was no mouth guard, both dogs were At Fault.