How do I make a GUI specific to one character?

I have a ScreenGui in StarterGui, and its Visible property is false by default. When an event is fired, I want it to be visible to the player that fired the event only. However, when the event fires it becomes visible to all players.

Should I be storing it in PlayerGui or something?

EDIT: This causes the gui to disappear on every player's screen. EM is off.

game.Workspace.StartLine.Touched:connect(function(p)
	if p.Parent.Name == "Car" then
		script.Parent.Visible = false
	end
end)

It's really confusing me because this code works for a different GUI.

local ReplicatedStorage = game:GetService("ReplicatedStorage")
local chooseCar = ReplicatedStorage:WaitForChild("chooseCar1")
script.Parent.Visible = true

script.Parent.MouseButton1Click:connect(function(p)
	script.Parent.Visible = false
	chooseCar:Fire()
end)