It is the individul. However, purchase and transfer records are kept at the Department of Public Safety. Recently, it probably works, (Computerized Instant Check at Gun stores...(The damned few remaining)) but ten years ago, it was allegedly a small room filled floor to ceiling with shoe boxes stuffed with "Green Cards".
So supposedly they know every piece of hardware a (legal) owner has. In this way, if a person ever has a restraining order against them, it's a "Christmas Shopping" orgy. The Sigs and 1911's would probably never be seen again, and the Brycos, Jennings, and Ravens would probably be returned someday. Maybe.