People who open your list's link see everything on it: its wishes, the date, which wishes are already taken and your name (not your photo), linked to your profile.
That's the list's name, description and cover, and every wish with its photos, price, shop, notes and priority, if you've set one. At the top it says "Wishlist by" and your name. Your photo and bio stay on your profile; the only people who see your photo on the list are those helping you run it.
They get a simpler page than you do: the Wishpicks logo, your name and the list, without the bars you use to add and organise wishes. It looks the same whether the list is set to Everyone or Anyone with the link. Who can open it at all is up to you: see "Who can see my wishlist?".
What can they do on my list?
They can reserve gifts and go to a wish's shop, but they can't add, edit, delete or reorder wishes, or change any setting. For what reserving looks like on their side, see "How do I reserve a gift?". To let someone help run the list, invite them: see "How do I invite someone to edit my list?".
Can I share a list without my name or address?
Your address shows only if you type it in, and a list shows your profile name, which can be a nickname. A list you made without an account says Anonymous until you move it into one.
To switch to a nickname, see "How do I edit my profile?". To get a gift sent to you without putting your address on the list, see "Does Wishpicks deliver the gift?".
What does the link preview show?
Paste the link into a chat and the preview shows the list's name, description and a picture of its first wishes. An Only me list's preview shows just its name, a general line about a private wishlist and a blurred picture with a lock, never the description or the wishes.
How can I see my list the way they do?
Open its link in a private browser window, where you aren't signed in.
On your list, tap Share (just the share icon on a phone), then Copy.
Open a private (incognito) window in your browser.
Paste the link. You'll see what anyone with the link sees, or, if the list is set to Only me, what someone who isn't invited sees.
Signed out, you'll also see which wishes are taken, just as they do, so skip this if you'd rather keep the surprise.
Tips
Holding a wish or two back for now? Move them to a second list set to Only me (see "Who can see my wishlist?"), then move them over when you're ready.
The date is optional, so leave it off a list of everyday wishes, or any list whose date you'd rather keep to yourself.
Send the link to yourself first, in the chat app you'll use, to check the preview before anyone else sees it.