Your list's Who can see it setting decides: Only me (private: you and people you invite), Anyone with the link (link-only, the default) or Everyone (public, on your Wishpicks profile).
Only me lets in just you and people who've accepted your invitation; a pending invitation isn't enough. The list shows "Private" with a lock under its name, and its card says "private" unless a date shows there.
Anyone with the link lets in whoever has the link, no account needed, and keeps the list off your profile. Its card carries no label.
Everyone lets in anyone with the link too, and also shows the list on your profile. For what your followers see, check "How do I follow someone?".
Wondering what people see once they're in? It's all in "What can people I share my list with see?".
How do I make my list private or public?
In your list's settings, tap Who can see it and pick Only me or Everyone; it saves straight away.
Open your list's settings with the sliders icon at the top.
Tap Who can see it.
Pick Only me to make it private, Everyone to make it public, or Anyone with the link in between.
You'll also find it in the Share window's Who can see it row, or by tapping Private on an Only me list. Without an account, Everyone is greyed out with "Needs an account".
After you pick Only me, people who've accepted an invitation still get in; to stop sharing with them too, see "How do I share my wishlist?".
Using a private list in a Secret Santa group opens it to anyone with the link, and it stays that way. Set it back to Only me and your group can't see it anymore: whoever drew you gets gift ideas instead, and the group's page offers Open it to anyone with the link to undo it.
Why does a list say No Access?
It's set to Only me, so only its owner and people who've accepted an invitation can open it. Everyone else sees "Owner has restricted access to this wishlist."
If it isn't your list and you've been invited, accept the invitation first: see "How do I join a list I was invited to?". Not invited yet? Ask the owner. If it's your list, switch it to Anyone with the link, or invite your friend (see "How do I invite someone to edit my list?"). A deleted list shows "Wishlist not found" instead.
Will my wishlist show up on Google?
It shouldn't: Wishpicks asks search engines to leave every wishlist page out of their results, whatever its visibility, but a link you post somewhere public can still be found. Inside Wishpicks, no setting puts your list in Explore, and search finds products, never people's lists. There's more on how Wishpicks protects your data on the Security page.
Tips
Anyone with the link suits most lists: friends open it straight from a chat, no account needed.
Want only certain people in? Set it to Only me and invite them as Viewers, so a forwarded link won't open it for anyone else.
Still putting a list together? Keep it on Only me, then switch to Anyone with the link when you're ready to send it.