Non-ownership of the dog, is a defense only to the strict liability provisions of MCLA §287.351. For example, if you are babysitting a dog, while its owner is out-of-town, and (heaven forbid) the dog attacks someone on your property, you cannot be sued under the strict liability provisions of the statute, nor can you sue the dog-sitter under the statute. However, you can sue and be sued, under a theory of common-law negligence, which is preserved in Michigan law under MCLA §287.288.
In the event the dog is being kept by a non-owner, there is nothing that prevents you from claiming damages from both the owner, on a strict liability theory, and against the dog-sitter, on a common-law negligence theory.