Not by default. Reveal reservations starts off to protect the surprise, so you can't see that anything on your wishlist was claimed. Turn it on and you'll see which items are taken.

You can switch it back off whenever you like.

Every time someone reserves a gift, you get an in-app "Gift reserved" notification. It names the wishlist and nothing else: not the item, not the person. If the Gift reservations notification setting is on, you may also get a reservation summary by email. Neither one spoils anything, since reservations stay hidden until you reveal them.

Revealing tells you an item is taken, but not always who took it. Whether a name shows depends on a second privacy setting.

Reserved and received aren't the same thing. The badge means someone plans to give that gift, not that it has arrived. Marking an item as actually received is a separate step you take yourself.