By default, no. Reservations are hidden from you, and the gift-giver stays anonymous. A name shows only if you do two things: reveal reservations, and set reservations to require registered users.

Reveal reservations is off to start with. Until you turn it on, you won't even see that items have been claimed.

Reservation privacy decides the rest. With Anonymous Reservations, the default, you see that an item is Reserved but never who reserved it. With Registered User Reservations, only signed-in people can reserve, and you can see the reserver's name and public profile.

So a name appears only when reservations are revealed and set to Registered User Reservations.

You can't see who bought a gift at all. Wishpicks isn't a shop and doesn't track purchases at the store, so there's no "bought" status, only reservation, which is intent. The store link on a wish just sends you to the merchant and records nothing.

A Reserved badge means someone plans to give that gift. It doesn't mean it has shipped or arrived.

One thing to expect if you switch on Registered User Reservations: the reserver's name is included when you export your wishlist.