As previous answers have already demonstrated, there's no consensus on this, even among "gun people" who ostensibly recognize and honor the Second Amendment. The Second Amendment doesn't say anything about having to provide documentation to purchase a firearm. In theory, then we should all be opposed to documentation of any kind to buy or sell a firearm.
But we don't live in a theoretical world, we live in the real world, and the real world is very litigious. So people want to protect their posterior anatomy, and rightly so.
I'm unfortunate enough to live in a state that doesn't allow face-to-face sales of firearms without calling the state police and getting their blessing, so I don't have to worry about what documentation I might or might not wish to see. It's all required by law, and rather than deal with it I would just go through an FFL with a consignment sale in the unlikely event that I might sell a firearm.
But I do think about what I would do if I lived in a free state. The libertarian in me says that, given my druthers, I would buy guns through channels that [legally] don't create paper trails. I'm not a prohibited person and I am legally allowed to own firearms. Therefore [in my mind], the government has no legitimate reason for knowing if, when, or how I purchase a firearm. Therefore, in fairness, I should accord the same philosophy to someone who might want to buy a gun from me. Federal law only requires that you cannot sell to someone whom you know or have reason to believe is prohibited ... and you can't sell handguns to residents of other states. The law does NOT require that you in any way verify that the buyer is prohibited -- the law says you can't sell if you know or have reason to believe that the buyer is prohibited.
So you could use a gut instinct litmus -- if the buyer shows up wearing a nice pair of Dockers and a Ralph Lauren polo shirt, isn't all decked out in tattoos and body piercings, and speaks comprehensible English, technically you don't have to ask anything. Or you could ask if he's prohibited, and if he says "Nope" then you're good to go.
On the other hand, if the buyer has dreadlocks, wall-to-wall tattoos, 25 pounds of gold in various body piercings, and talks only in monosyllabic ghetto speak, then you might "have reason to believe" he's prohibited, and ask for some sort of documentation.
Another level of wrinkle, in my personal [theoretical] analysis is: where did I get the firearm I'm selling? I'd like to buy guns off-paper, so in fairness I should respect that other people might also want to buy off-paper. If the laws of the jurisdiction allow for that, maybe that's okay ... IF I bought the gun off-paper. But if I bought it through an FFL or any other avenue that left a paper trail leading to me, then it's only reasonable that I also want a paper trail leading away from me if/when I sell a gun that I bought on-paper. Bills of sale are not unbiased. I can make up a bill of sale in a fictitious name, and sign a fictitious signature. IMHO a bill of sale that I created proves nothing. If I want a paper trail to verify that I sold that gun, I'm better of handing it over to an FFL, who will enter it into his bound book, for consignment sale.
The Cliff's Notes version of the above, I guess, is that there's no simple answer to the question.