The rules text could stand to be more explicit about this, I think. You can't note a creature type that's already been noted for that copy of Long List of the Ents. You can note a creature type that's already been noted for another Long List, regardless of who controls it.

So say you control four copies of Long List. You can note "Treefolk" one time for each of them, but you can't note "Treefolk" more than once for any of them. Same if every player controls a Long List. Everyone can note "Treefolk" once for their own copy.