To let someone edit your wishlist, invite them by email from the wishlist's access settings and give them an access level — they then accept the invite and sign in with that same email. A plain shared link only lets people view, never edit.
Steps
Open the wishlist you want to share for editing.
Go to its access (sharing) settings.
Enter the email address of the person you want to add. (Editing is granted by email, not by username or search.)
Choose an access level. The invite offers three: Full Access, Contributor, and Viewer. For someone who should edit, pick one of the first two:
Contributor — they can add, edit, and manage wishes, but can't rename or delete the wishlist.
Full Access — everything a Contributor can do, plus changing the wishlist's settings and deleting it.
(Viewer is read-only — choose it only if you want them to view and reserve, not edit.)
Send the invite. Wishpicks emails the person an invitation with a link to accept it. If it doesn't reach them, open the person's entry in your access list and choose Resend invitation to send it again.
They accept the invite and sign in with the same email you invited. After that, they can edit the wishlist.
A few details
The invited person must (1) accept the invite, (2) be signed in to a Wishpicks account, and (3) use the exact email you invited. If any of these is missing, they won't be able to edit — see "Why can't someone edit my wishlist?"
Wishpicks sign-in is via Google, Facebook, or an emailed magic link. If your invitee is new, they can register with the invited email first, then accept.
You do not need to make the wishlist private to add an editor. Visibility controls who can view; the access level controls who can edit. They're independent.
A Contributor can only edit or remove the wishes they added themselves, unless you give them a higher level.
Invites are generous, but not unlimited. You can put up to 50 email addresses into a single invite and send at most 100 invites a day across all your wishlists, and the same email can be invited to the same wishlist at most twice in 5 minutes and three times a day — a guard against accidental spamming. Hit a limit and you'll just need to wait a bit before trying again. Nothing expires, though: the accept link stays valid until the invite is accepted or you remove the person's access, so there's no rush for the invitee.
You can also invite more than one person overall — see "Can several people manage one wishlist together?" Not sure which level to pick? See "What is the difference between viewing and editing access?"