Every wishlist has one of three visibility settings: Public, Anyone with the link, or Invite only. They control who can view it. New wishlists start on Anyone with the link.
You change this under the Share tab in your wishlist settings, in the section for who has access.
The three settings
Public makes the wishlist viewable by anyone, and it also shows on your public profile. You need to be signed in to pick this one.
Anyone with the link is what new wishlists start with. Everyone you send the link to can view it, and it stays off your profile.
Set a list to Invite only and only you and the people you've invited by email can open it. Everyone else gets a "No Access" page.
Good to know
Visibility controls viewing, not editing. Letting someone edit is a separate email invite, and it works the same whether the wishlist is public or private. See "What's the difference between an editor, a guest, and someone reserving a gift?"
To share an Invite only wishlist, a plain link won't do. Invite people by email address instead. The Share screen reminds you: turn off Invite only if you want a link to work.
Guests can reserve gifts on any wishlist they can view, unless you've changed your reservation setting. See "How do I make sure my friends can reserve a gift?"
One habit worth keeping: don't put a home address on a wishlist other people can open. Wishpicks has no delivery-address field. The gift-giver buys at the store and works out delivery with you directly, or you send the details privately.