In the TV show Quantum Leap Sam Beckett would travel history and "right" wrongs, temporarily displacing someone native to that timeline in the process. Would the memories of what Sam did transfer to the person he displaced upon his leaping out, or would they wind up with amnesia for the time Sam was there?
It appears that there is only one episode where a Leapee (Frankie) is ever shown after returning to their time post-leap: "Double Identity". He appears to have amnesia, not having any memory of what happened over the previous day or his time in the Waiting Room while Sam was in his place.
You can actually watch the episode on Hulu.
Sam would switch with the "leapee" and attempt to fix whatever they'd done to muck up their timeline. Generally this would be to affect a single bad decision rather than to make any large-scale lifestyle choices on their behalf. This means that once they'd returned, they would no longer be in a position to make that 'bad' choice and simply carry on with their lives as before.
Per the Quantum Leap FAQ
Q. What does the leapee remember about his experience after he returns?
This is also not known. The only time we've seen this occur was in the episode "Double Identity," where Sam leaped to replace another body and the original host returned. He APPEARED to have no memory of anything after he was leaped into. It has been stated that the leapee, while in Sam's body back in the Waiting Room, has a 'swiss-cheesed' memory, much like Sam received upon his initial leap. Because of the ultramodern hospital-like atmosphere of the waiting room, many of the leapees believe they have been abducted by aliens. Deborah Pratt says that as the leapees return, they pick up some of Sam's memories of what happened, but they believe the events happened to them.