By default, no — reservations are hidden from you and the gift-giver stays anonymous. You'll only see who reserved an item if you both reveal reservations and set reservations to require registered users.
Two separate privacy layers have to line up before a name shows.
How it works
Reveal reservations (off by default). Until you turn this on, you don't even see that items were reserved. See "Can I see if someone reserved my gift?"
Reservation privacy. Even after revealing, whether you see a name depends on this setting:
Anonymous Reservations (the default) — you see that an item is Reserved, but never who reserved it.
Registered User Reservations — only signed-in users 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.
Good to know
You can't see who bought a gift. Wishpicks is not a shop and doesn't track purchases at the store, so there's no "bought" or "purchased" status — only reservation, which is intent. The store link on a wish is just an outbound link and records nothing.
Reserved means intent, not delivery. A Reserved badge tells you someone plans to give that gift, not that it has shipped or arrived.
Named reservations also show up in exports. If you've set Registered User Reservations and export your wishlist, the reserver's name is included in the export file.
The default protects everyone's surprise. Most owners leave reservations hidden and anonymous on purpose, so the gift stays a surprise and gift-givers stay private.