A reservation, not a purchase
On a shared wishlist, any signed-in guest can reserve a gift. Other guests immediately see that it is taken, along with the name and avatar of the person who reserved it. No one needs to ask the group chat whether the coffee grinder is handled.
A reservation is not a payment. It is a lightweight promise that keeps the list honest.
The surprise stays intact
The person who owns the wishlist never sees who reserved what. Their list stays a list of possibilities. The guests coordinate among themselves, and the surprise survives the logistics.
To make this meaningful, the server omits reservation information entirely from the owner's authenticated view, rather than hiding it with a blur or a CSS trick.