Nordyke, it now appears, might help with the answer. Last week, four preeminent legal scholars—Michael Kent Curtis, Richard Aynes, Michael Lawrence, and William W. Van Alstyne—filed a friend of the court brief arguing that the 14th Amendment "and specifically its privileges or immunities clause were designed to forbid states from abridging fundamental rights of citizens, including those rights in the Bill of Rights." The Second Amendment, of course, is right there on that list.
For its part, Alameda County currently maintains that, "the Second Amendment constrains only Congress," while the amendment's core purpose of "self-preservation...is best advanced through the establishment and exercise of the police power." Which means that while Congress may not legally disarm the American people, state and local governments may.