Good point, what do you think the ratio of private transfers to background checks? (I mean the private transfers that don’t go through the paperwork, since some private transfers do).
Good question; I don't know of any hard data on the subject. About a quarter of my firearms (rough estimate) have been purchased without the need for a background check. When Colorado passed a gun show background check bill about 10 years ago, I started undergoing many more checks. Earlier this year, Colorado passed a bill requiring background checks for any private transfer, so the only guns I acquire that won't need a check are those I get via my Curio & Relic FFL.
Just as a guess, I would estimate that for every gun which is sold via a background check, another is sold privately without one.