Once you've signed in, open People with access in your list's or collection's Settings, type their email, pick Contributor or Full Access and tap Send invites.
There's no special shared-list type: any list can take a partner or the whole family. Sending the link never lets anyone edit; only an invitation does, and you don't need to change who can see the list first.
Steps
Open your list, tap Settings (the sliders icon by the share button), then People with access. Who has access in the Share window opens it too.
Type their address in Invite by email, with commas between several.
Under Access level, pick Contributor or Full Access to let them edit, or keep Viewer to let them see the list, even one set to Only me. The line underneath explains each one.
Tap Send invites. Each person gets an invitation email, and their row appears below, marked "pending" until they join.
"This user is already invited" means they're on the list. If Wishpicks says you've reached a limit or asks you to try again later, wait before sending more invites.
Why does it say I need an account to invite people?
Inviting people needs an account, and you made this list without one: tap Create account or Sign in right there, then send your invites. "Do I need an account?" shows how to move it to your account.
They didn't get the invitation email. What now?
Ask them to check spam, then tap ⋮ on their row in People with access and choose Resend invitation. It arrives even if they've turned off other Wishpicks emails. Typo? Tap ⋮ → Remove and invite the right address. With an account on that address, they can join without the email: see "How do I join a list I was invited to?".
Why can't the person I invited edit my list?
Usually they haven't joined yet or their level doesn't let them edit; their row in People with access shows which. You aren't told when someone joins, but no label under their name means they have.
"pending": they haven't joined, or signed in with another address. Invite the address they use.
"rejected": they declined or left. You can invite them again.
Viewer: raise their level to Contributor or Full Access.
Contributor: some changes need Full Access; see "What can each access level do?".
No label: they may be signed out, or in another account, on that device. Ask them to sign in with the invited address.
How do I remove someone or change what they can do?
In People with access, tap the level by their name to pick another, or tap ⋮ → Revoke access (Remove if they haven't joined) to take it away; neither change emails them. "Access changed" confirms a new level, and wishes they added stay. On a list open to anyone with the link, they can still view it like any visitor: see "Who can see my wishlist?".
Tips
Give people a heads-up: the invitation comes from Wishpicks, not from you.
Inviting people at different levels? Send one invite per level: everyone in a send gets the same one.
When the occasion's over, drop helpers to Viewer or revoke their access.