Does a CharacterAdded function run separately for each player?

UPDATE: It works (LocalScript) now, which funnily. you can find in jav's answer, however, after the first time I change my class to an accepted Class (i.e. zombie, nighthunter, etc.), it displays a GUI, which respawns when you click a button. When I click it, it puts the GUI back. I've tried to work around this with the 'x' variable, but it still doesn't seem to work.

Hello. I've been writing a script, where, if a character respawns as certain 'classes' (a StringValue), the script will parent a Gui to them when they respawn.

I've shuffled between using Server-side scripts and LocalScripts, but LocalScripts don't seem to do it, correct me if I'm wrong; and the Server-side scripts is what I had a question about.

Does a CharacterAdded function run separately for each player? Meaning if I declared a variable 'x' (somewhat of a debounce variable), would that 'x' be tagged to that one player respawning, or would the 'x' be the same across all respawns?

Here's the code:

game.Players.PlayerAdded:connect(function(Player)
    x = "Stop"
    Player.CharacterAdded:connect(function(Character)
        wait()

        if x == "Go" then



                if Player.Class.Value == "Zombie" or Player.Class.Value == "Nighthunter" or Player.Class.Value == "Ogre" or Player.Class.Value == "Thing" or Player.Class.Value == "Imp" or Player.Class.Value == "Golem" or Player.Class.Value == "Savage" then
                    wait()
                    _G.StopPlr(Player)
                    if Player.PlayerGui:FindFirstChild("MonsterSelect") == nil then
                        local ms = game.ServerStorage.MonsterSelect:clone()
                        ms.Parent = Player.PlayerGui
                        x = "Stop"
                        else print("Defaulted to Zombie Class for ".. Player.Name)
                    end
                    else print("ERRRRRR")
                end

            --controls choose monster GUI
            else x = "Go"
        end

    end)
end)

Notice how I continually reference 'x' in the script? If someone respawns and x is 'Go', and then another person respawns, will that same x change, or will it be an entirely different x altogether? Sorry if that's a bit too confusing.

Thanks for reading; all replies are appreciated!