How do I make a Billboard Gui only visible to the local player?

If you have the BillboardGui inside of StarterGui, you can set its Adornee to the object you want it to appear for, rather than putting it inside the object itself. If you set the Adornee through a LocalScript, it should only be visible for that player. Theoretically, all of your Surface, Billboard and Screen guis should be inside the player's gui, never actually inside the part itself (at least with FilteringEnabled).

Hope this helps!