1. the first concern that raises antenna is whether background checks will be used to make de facto registration lists. Some people don't want the govt knowing what they have, so that the govt can't then target them specifically with increasing measures. The 2nd amendment is supposed to enable a citizen militia to preserve a free state, in the event the govt attempts to revert from democracy to tyranny, and registration or other means to make a "firearms owners census" undercut the ability to preserve arms outside the view of govt. That said, it's not clear that individual check results would be preserved, currently they are not for commercial sales but the anti-gunners don't like that.
2. which ties in with the question, how would it work? How do I prove that I did a background check on a person before selling them a gun? Either I or the govt needs to keep a record to prove it. Gun shops fill out a form and are trusted to do so but the results aren't kept. Will the govt place the same trust in private sales, i.e. not require any verification of actual compliance?
3. Or will the govt require private gun sales to occur at a gun store with verification by the store employee, which would add service charges and also allow the govt to collect taxes on the sale, and make it much more inconvenient for the buyer/seller. Plus potentially requiring a waiting period after payment for transfer of the gun. Also currently private sales on dealer property is a felony so that would have to be changed.
It could potentially be worked out to not incur significant additional burdens on private citizen transactions. But that seems unlikely, and what is especially irksome is that the burdens are being imposed to solve an essentially nonexistent problem. There is not much evidence that legitimate private sales are responsible for a significant amount of criminal gun usage. Not in any of the recent mass shootings, and most guns used in crimes come from a very small set of dealers who specialize in selling to gang members. Also there are some individuals who specialize in "straw man" purchases, they buy dozens of guns a month and then sell them privately to gangbangers - but this is already illegal. Instituting a private sale background check won't prevent that, unless it is so incredibly burdensome and thorough that it adds all of the problems noted above to legit private sales. Then it might stop some straw man sales but . . .those are already illegal without the universal background check requirement and the govt is already able to enforce those laws!!! So it's creating a huge burden to create an unneccessary answer to a problem that already has an answer, that's just not being utilized.