Invite them by email from the wishlist's access settings and give them an access level. They accept, sign in with that same email, and then they can edit. A plain shared link only lets people view.
Steps
Open the wishlist you want to share for editing.
Go to its access (sharing) settings.
Type in the email address of the person you want to add. Editing is granted by email, not by username or search.
Choose an access level. You'll see three: Full Access, Contributor, and Viewer. For someone who should edit, pick Contributor (add, edit and manage wishes) or Full Access (all of that, plus changing the wishlist's settings and deleting it). Viewer is read-only.
Send the invite. We email the person an invitation with a link to accept. If it doesn't reach them, open their entry in your access list and choose Resend invitation.
They accept and sign in with the same email you invited. From then on they can edit.
Good to know
Editing needs all three of these: an accepted invite, a signed-in Wishpicks account, and the exact email you invited. Miss one and nothing happens. See "Why can't someone edit my wishlist?"
Signing in works through Google, Facebook, or an emailed magic link. Someone new can register with the invited email first, then accept.
You don't have to make the wishlist private to add an editor. Visibility decides who can view; the access level decides who can edit.
A Contributor can only edit or remove the wishes they added themselves, unless you give them a higher level.
You can put up to 50 email addresses into a single invite. To stop the invite form being used for spam, there's a limit on how many invites go out in a short time. If you hit it, wait a few minutes and try again. Invites don't expire, so the accept link stays good until it's used or you remove the person's access.
Related
"Can several people manage one wishlist together?"
"What is the difference between viewing and editing access?"