By default, no — you can't see that anything on your wishlist was reserved, because the Reveal reservations setting is off to protect the surprise. You can turn it on, and then you'll see which items have been claimed.
This is on purpose. Most people want gifts to stay a surprise, so Wishpicks hides reservation activity from the owner until you reveal it yourself.
Steps
Open your wishlist as the owner.
Find the Reveal reservations setting.
Turn it on. You'll see a "Surprise Spoiler Alert!" warning first, because this can reveal which gifts you're about to receive.
Once revealed, reserved items show a Reserved badge.
What to expect
You get a quiet heads-up either way. Each time someone reserves a gift, you receive an in-app "Gift reserved" notification that names the wishlist — but not the specific item or the person — so it won't spoil the surprise. You can also receive an email summary if the Gift reservations notification setting is on; it still keeps the surprise, since reservations stay hidden until you reveal them.
Revealing tells you an item is taken, not always who took it. Whether you see the gift-giver's name depends on a separate privacy setting. See "Can I see who reserved or bought a gift?"
Reserved is not the same as received. A Reserved badge means someone intends to give that gift, not that it has arrived. Marking an item as actually fulfilled is a separate step you do yourself.
You can also turn the setting back off. Revealing reservations is your choice; you stay in control of how much you want to know.