If I give every player a GUI in which they can select their spot, how do I make it show other players which spot has already been picked?
Do you use a remote event to fire to the server and then another to fire back at all clients? 
That is what I would do. I would fire to the server and then fire to all clients because I don’t think it is possible to change other peoples UI’s on the client side of another client.
I just made up a quick sample of a GUI where a player can select their spot if it’s not picked, and it will tell them if it’s picked or not. You can take a look at it, and try to revamp it into your game: SpotsEdited5.rbxl (27.4 KB)