Our pitfall in selling privately was (perhaps) that our buyer lost his mortgage. We knew that he was fiddling the system, pretending it was a residential buy rather than a buy-to-let, but decided that if the Building Soc. was willing to lend the money, we'd let it go. We also wondered if he could really make the payments -- get the rent to cover, but again, didn't seem like our problem... It was after we exchanged contracts that the buyer's mortgage advisor got caught out, and buyer lost his mortgage. Buyer didn't qualify to get a legit mortgage.
We were naiive in thinking it "wasn't our problem" that we knew the buyer's mortgage wasn't legit, and I suspect the Estate Agents might have caught it because they seem very good (we are now selling through an agent) at checking financing.
BUT, a lot of people in our town (presumably almost all selling through agents) were caught out like us, where mortgage offers from a particular local Building Soc branch were no good -- it was a our own local version of the Self-Cert scandal. So I don't know if selling via an agent would have really protected us or not.
Lesson learnt: I would be blunt about asking about buyer's finances in the future, if selling privately. We were lucky we were planning to rent and hadn't even started looking yet, when our buyer fell through.