To make a wishlist private, open its Share settings and switch the visibility to Invite only. After that, only you and people you specifically invite can see it — everyone else gets a "No Access" page.
Steps
Open the wishlist you want to make private.
Open its settings and go to the Share tab.
Set the visibility to Invite only.
Save.
A few details
Once a list is Invite only, a plain link no longer works for other people. To let a specific person view a private list, invite them by email — see "How to share a private wishlist".
New wishlists are Anyone with the link by default, so they are not private until you change the setting.
Making a list private doesn't change who can edit it. Editing access is controlled separately by email invites and access levels (Owner, Full Access, Contributor, Viewer), not by visibility. So you don't need to make a wishlist private to add collaborators — a public list can still have people who help manage it.
To go the other way and let everyone see a list, see "How to make a wishlist public".
Related
Who can see my wishlist?
How to share a private wishlist
How to make a wishlist public
What information is visible to guests?