By default, you don't even see that anything was reserved on your own list — so the surprise stays safe. You only see reservations after you turn on Reveal reservations yourself, and whether you see who reserved depends on your reservation-privacy setting.
How the surprise stays protected
Reveal reservations is off by default. With it off, you don't see that anyone reserved anything, so nothing spoils the surprise — you don't have to change a thing. When you turn it on, Wishpicks warns you first with a "Surprise Spoiler Alert!" so you can back out, and reserved items then show a Reserved badge so you can see what's already covered.
Whether you see who reserved
Even after you reveal reservations, seeing the gifter's name depends on your reservation-privacy setting:
Anonymous Reservations (the default) — You see that an item is reserved, but not who reserved it.
Registered User Reservations — You see the signed-in reserver's name and profile.
So out of the box, gifts stay a surprise and gifters stay anonymous. You opt in to each one separately.
What to expect
You still get a quiet, in-app Gift reserved notification when someone reserves an item. It names the wishlist, not the item or the person, so it won't spoil the surprise. If your Gift reservations notification setting is on, Wishpicks may also email you a periodic reservation summary. The surprise stays protected either way — what's reserved stays hidden until you choose to reveal it.
Reserving a gift isn't the same as buying it. A reservation marks intent only; you mark an item as gotten separately. See the gift-reservation articles for the full picture.
Related
Can I hide reserved gifts from myself?
Can I see who reserved or bought a gift?
What information is visible to guests?