Currently we have no gun registration. Much impending legislation is trying to change that, and as it's been a D initiative for a long time we have to expect they'll ram it through.

Currently they (supposedly) have to go to a FFL (like a gun store) and check through whatever records they have there. Which they could do. So gun confiscation would be fairly easy in large numbers, at least as far as locating many goes. This doesn't help them much in terms of finding any specific gun. In practice nothing stops them from maintaining all info of a background check, which includes the type of gun. So giving feds the serial # tells them you haven't lost it in a boating accident, but maintain it proudly.