What legal responsibilities does a listed inventor have?

The inventors responsibility, as an inventor, is to truthfully sign the declaration regarding being a true inventor and fulfilling the duty of disclosure and candor to the USPTO during the prosecution of the patent.

It would be the marketer/producer that would have any liability for a shipping product based on the patent. A patent doesn't confer the positive rights do anything in particular to the owner of the patent. It only provides the negative right to try to stop someone else from practicing the claims.

It is well known that the USPTO does no research or experiment to determine is something works. Only something very, very impossible on its face - like perpetual motion - gets rejected as "not useful". In fact you can get a patent on something that looks like an attempt at perpetual motion if you say the utility is that it is entertaining to watch it spin around for a long time.

The specification of a patent can/must make assertions about the usefulness of the invention and "patented" as a marketing hype is often done, but as @EricShain points out, it is a product from some manufacturer/marketer that is hyped to consumers, not the idea of the invention itself.