The registration I have been told by HPD never leaves your name. Even of you transfer it to someone else. You are 100% responsible to know who you sold it to.
Is HPD claiming their jurisdiction applies to a private sale in New Mexico? Or just that no matter what you do wherever with a firearm registered to you, if they ever come to your door with a warrant to confiscate it, you will be arrested, charged and prosecuted unless you produce it?
How much information do you have to provide to HPD about the person you sold your firearm to in New Mexico? Name, address, social security number, photocopy of driver's license, photocopy of passport, color photo, fingerprints, bill of sale, video recording of entire transaction (as is mandatory in the new Chicago gun store law)?
Seems like the only real answer is to have a national registry of all firearms transfers, conducted through FFLs or some other government agency, then no matter who sold/transferred whichever gun to whomever wherever, the government would have a complete and total record of all that information. Well, except for criminals who steal their guns or acquire them via the black market., and that's such a relatively small number it's hardly worth bothering about compared to the law-abiding gun owners who are the real problem. [/sarcasm]