Generally speaking, the leapee doesn't remember anything of their experience in the waiting room.
The only time we directly see a leapee after a leap occurs in the episode "Double Identity." In this episode, Sam first leaps into Frankie, a young Mafia hitman having an affair with Teresa, the lover of the crime boss, Don Geno. At the end of the episode, the Quantum Leap project executes an attempt to get him back (inadvertently causing a famous historical blackout) and Sam leaps without accomplishing his mission... into the body of Don Gino in the same room seconds later. So, we see the immediate reaction.
(excuse some of the iffy attribution, I found a transcript of the scene but it didn't include any of the names and I don't have the episode on hand to watch)
T: Frankie, are you all right, hon?
F: Teresa, hey, you talk about your earth moving. -
Sam: Oh, God.
F: Don Geno!
Sam: What's happening?
F: Happening? Nothings happening. We was, uh We was looking for a suitcase. Uh, Primo wanted us to get a suitcase for the honeymoon. Were we hit by lightning?
Sam: Something like that.
F: It's night time. What happened to the wedding?
T: That was yesterday.
F: Yesterday?
Sam or T: You got a concussion. Lightning can do that.
Sam (possible voiceover): Either Ziggy's blown it again, or whoever's bouncing me around in time isn't letting me off the hook. I was put here to do something, and now that I was Don Geno, I finally knew what it was.
Sam: Teresa, do you love Frankie?
T: Yeah.
Sam: And Frankie, do you love Teresa?
F: Love Teresa? Godfather, anyone even looks at Teresa, it's...
Sam: What if it wasn't Do you love her?
F: With all due respect, Godfather, I love her so much it hurts.
T: Oh, Frankie.
Sam: Terrific. What's best for all of us is for you two to be together. The trouble is as soon as I put you together, I'm outta here, the old Gino is back, and you end up like Jimmy Hoffa.
F: Running the Teamsters?
So he has to make sure the whole city knows he approves of the union in a way Don Geno can't go back on later.
However, we see that Frankie doesn't remember ANY of his time either in the waiting room or while Sam is leaped in. As far as he's concerned, it's still the day of the wedding that Sam leaped into. We don't know for sure if this is normal (they were trying an experiment at the project to get Sam back, and that might have messed with normal procedures), but we do know that Sam expected Don Geno to not remember or care that he gave permission if he didn't do it publicly, and Sam's leaping has given him a "swiss cheese" memory that, it would stand to reason, also affects those he leaps into, at least for the relevant period.
We also have the note that Sam has, occasionally, leaped into famous people, who presumably never spoke or wrote of their waiting room experiences. Granted, those experiences would be considered the equivalent of an alien abduction, but more importantly they've never had notable memory losses, either. That suggests that it's POSSIBLE the leapee remembers some of what Sam did while in their body.
There is some justification within the rules set up by canon for the leapee to remember, though. They frequently talk about the leapee and Sam exchanging "neurons and mesons" during the process. Usually this gives Sam bits of the leapee's personality (notably in the episode where he leapt into Lee Harvey Oswald and he got more of Oswald's neurons and mesons than usual), but it could also work in reverse, giving the leapee some of Sam's memories of the few days he or she was not in control, perhaps with some confusion and fuzziness at least at first but not a complete gap.
Most likely, because it involves humans and humans are widely variable, all scenarios happen sometimes: Sometimes the leapee has no memory of the hours or days Sam was there, sometimes they remember some of what Sam did in their place (albeit probably less well than they otherwise do), and sometimes they have weird memories of being interviewed in a waiting room and seeing someone else's face in a reflection (which they almost certainly assume was some weird dream). It all depends on what God, or Time, or Fate wants them to remember.
