Yes, and it's already the case. Reserved gifts are hidden from you by default, so nothing spoils the surprise. You'll only see what's been reserved if you switch on Reveal reservations yourself.

With Reveal reservations off, your own list looks exactly the same whether or not people have reserved things. You don't have to do anything to keep it that way.

Turn it on and Wishpicks warns you first with a "Surprise Spoiler Alert!", so you can back out. After that, reserved items carry a Reserved badge and you can see what's already covered. Changed your mind later? Switch Reveal reservations back off.

Revealing reservations doesn't necessarily tell you who reserved something. That's a separate setting. Anonymous Reservations, the default, hides the gifter; Registered User Reservations shows the signed-in reserver's name.

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

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