A Viewer sees the list and can reserve, a Contributor adds wishes and edits their own, and Full Access changes any wish and the settings but can't invite people.
Viewer: sees every wish and can reserve, like any visitor, but also has the list in their Wishes → Lists and can open it even when it's set to Only me.
Contributor: adds wishes, and can edit, delete or mark fulfilled the ones they added, but not anyone else's. They can't change settings.
Full Access: adds, edits, deletes and marks fulfilled any wish, and changes the name, description, cover, who can see it, who can reserve and the date. They can delete the whole list too, but can't invite people.
Owner: you. Only you can invite people, change their level or take their access away.
On your list, a wish someone else added shows "Added by [name]". Who sees reservations differs by level too: see "Can I see what's reserved on my list?".
What can I do on a list I was invited to?
It depends on the level the owner gave you, and the Shared with you badge under the list's title shows which. On a computer, point at the badge and its tooltip says what you can do; on a phone, tap it to see "Your access", then tap Cancel to stay on the list.
With Contributor or Full Access, the list opens with your usual Wishpicks menu around it, the bar for adding wishes and the owner's name under the title. As a Viewer, you see it the way any visitor does.
As a Contributor, you can change the wishes marked "Added by me". Open a wish someone else added and you'll see "[name] added this wish, so only they and the list's owner can change it." People with Full Access can change it too. On the wishes you can change, Fulfill takes the place of Reserve: see "How do I mark a wish as received?". Not joined yet? See "How do I join a list I was invited to?".
Can I invite more people to a list I'm on?
No. Ask the owner: only they can invite people or change anyone's access. With Full Access you'll see People with access, and while nobody else is on the list, it says "Only the owner can invite people. Nobody else is on this list yet."
Tips
Save Full Access for someone you'd trust with the whole list, like a partner, since they can delete it.
For family or friends adding their own ideas, pick Contributor: they can't change anyone else's wishes.
If they might want to reserve a gift for you, keep them a Viewer: "How do reservations work?" explains who can reserve.
Not sure? Start lower: you can raise their level any time from their row in People with access (see "How do I invite someone to edit my list?").