The 'This Is My' segment HAS to be at least partially set up...
Now what I mean by that is, it wouldn't surprise me at all if each of the panelists were asked/told just prior to the show to already come up with their excuse for how they know the person... well, the 2 people lying, that is. Only because, Rob Brydon ALWAYS just immediately has little quippy things to call the person based on the panelist's initial answer, before they then turn it over to the other side to begin asking for details. I love Brydon, but I just don't think he can that often come up so quickly with the perfect little 5-6 word title for what that person is.
But pretty much everything else with the show, I believe it is legit, the panelists that are lying indeed are seeing the lie for the first time when they read it aloud (unlike something like "Whose Line Is It?" where so much of that show is set up beforehand not all genuinely improv'd).