I’m pretty sure they’ve built relationships over time with a lot of these actors – Lorne, or his people, probably call up Alec (or Melissa McCarthy, or whomever, or their agents) and offer them the gig. If they’re in NY at the time, or if they’re excited enough about doing it, they probably say “sure”.
In the case of Alec Baldwin playing Trump, it was Tina Fey who first suggested it to Lorne Michaels (cite). Baldwin has hosted SNL numerous times (and of course costarred with Fey in 30 Rock), so he’s no stranger to the show.
In the case of Melissa McCarthy, she started out in improv as a member of The Groundlings so doing impressions is right in her wheelhouse. The Groundlings and Second City account for most of the performers on SNL so I’m sure they were all very aware of her skill even before she hit it big.
In the case of Tina Fey playing Sarah Palin, Lorne Michaels admitted that his initial idea was to have current cast member Kristen Wiig play the part. On the way to the studio, his doorman remarked how much Palin looked like Fey. By the time he got to his office several friends and SNL staff had remarked on the resemblance, and Michaels decided to go with it.
Familiarity has a lot to do with it. The writers and cast members have all worked with the extra guests before and know they’ll fit right in. I remember reading the reason why SNL chose Buck Henry to host the season finale so often is because they wanted as few problems as possible and he was the easiest to get along with.
Affleck was in fact the host of that episode. During the pitch meetings earlier in the week, the writers will look at who the host does impersonations of or who the host looks like or sounds like and write sketches to highlight those.
Using a woman, even one as great and funny as Melissa McCarthy, to be the male press secretary seems like it would’ve involved some discussion or something. I don’t see them doing a “hey Fey = Palin!” on-the-spot observation.
I’d love to hear how they made the connection and got the gig. She killed.
Well, she’s known for doing “belligerent, self-serious doofus” roles skillfully (her character in Bridesmaids was sort of along those lines). So that would naturally bring Spicer to mind.
By the way, there are indications that Trump was freaked by the idea of a woman playing Spicer–enough so that the very fact of his having been played by a woman might lead Trump to give him the heave-ho:
Yeah, my first thought on hearing about this ‘Trump freaks out over Sean Spicer being played by a woman’ story was: recast everybody. Kellyanne Conway should be played by one of the men, and Bannon and Flynn definitely should be played by women. (My guess is that Trump is pleased and flattered, rather than annoyed, by having Bannon portrayed as the Grim Reaper, because the Grim Reaper represents power.)