With the new Steam beta update you can now mark obtained games as private, but not those on the whishlist, obviously because you don't have them yet.
I saw an idea in another post to mark individual games on the whislist as private.
My idea is to make multiple wishlists, one public and one private like in Itch.io
I think my idea is more developer friendly because it is simply creating a hidden list and there would be no need to modify anything else.