As far as I know, there is nothing prohibiting private citizens from going to an FFL and making a transfer, thus gaining (indirect) access to the NICS system. Accordingly, if a seller would like to take the extra step to ensure that he is not selling to a prohibited person, he is welcome to so.
Opening NICS up to private citizens opens the door to more incremental encroachments on our 2A rights. Here's one path that I can imagine, just off the top of my head:
1) Open NICS up to private citizens.
2) Require that all transfers go through NICS, on the theory that "the system is already open to the public, so there's no inconvenience in making folks use it. After all, it's for the children."
3) Learn that requiring all tranfers go through NICS didn't reduce crime, because it's darn near impossible to prosecute anyone for failing to use system without full gun registration.
4) Implement full gun registration.
5) Discover that convicted felons or other prohibited persons cannot be prosecuted for failing to register firearms that they should not have had in the first place. See Haynes v. U.S., 88 S.Ct. 722 (1968).
6) Figure that since registration didn't work, clearly the gov't didn't "go far enough" on gun control.
7) Decide on the next step . . . .
I am not in favor of opening it to the public.
I'm a lawyer, but I'm not your lawyer. If you need some honest-to-goodness legal advice, go buy some.