@Pinkbonbon I think you need 10....?
But, when someone's at my campsite that I want you have to schmooze them a bit by playing games and getting the card symbol/colours correct when they play a game to decide whether they should move to your island or not...
If your full, they'll go and talk to Tom and tell you that 'X' is thinking about leaving, and ask you if they should go and ask them to swap (This is random, but I usually find it's someone you've not talked to in a while or as often as others...)
I wish you did get to choose though! I spent forever trying to get one to move (kept getting the card wrong!) But didn't want to lose Marina!