Open the wishlist someone shared with you, tap Reserve on the wish, and confirm. That flags the item as taken by you, so nobody else picks the same thing.

There's no separate "I'll buy this" button. Reserve is the signal. Buying happens afterwards, at the store.

You may not need to sign in. If the owner allows Anonymous Reservations, you can reserve as a guest. If they've set Registered User Reservations, you'll be asked to sign in first.

If someone got there before you, the wish shows as taken and you can't reserve it until they cancel.

The owner usually can't see any of this. Reveal reservations is off by default, so the surprise holds. Reserving doesn't mark the wish as received either; the owner does that later, on their side.

Reserving and buying stay separate. Reserving holds the item on Wishpicks so gift-givers don't overlap, and the purchase happens at the external store. Wishpicks isn't a shop and never takes payment for a gift. There's no delivery-address field either, so sort out where to send it with the recipient directly, off Wishpicks.