By default you don't even see that something was reserved on your own list, so the surprise holds. Reservations only become visible once you turn on Reveal reservations yourself. Whether you then see who reserved is a second, separate setting.

Reveal reservations is off out of the box. With it off, your list looks unchanged no matter how many people have reserved things, so there's nothing for you to do.

If you turn it on, Wishpicks stops you with a "Surprise Spoiler Alert!" first. After that, reserved items show a Reserved badge and you can see what's already covered.

Revealing reservations shows you what is taken. Seeing the gifter's name depends on your reservation-privacy setting. Anonymous Reservations, the default, tells you an item is reserved but not by whom. Registered User Reservations shows the signed-in reserver's name and profile.

So on the settings you start with, gifts stay a surprise and gifters stay anonymous. You opt into each one separately.

You do get a quiet in-app Gift reserved notification when someone reserves an item. It names the wishlist and nothing more, so it can't give away the gift or the person. If your Gift reservations notification setting is on, you may also get a reservation summary by email now and then.

Reserving isn't buying. A reservation only marks that someone plans to give the item, and you mark it as gotten separately. See "Does reserving a gift mean I've bought it?"