Not by default: your Wishpicks reservation is anonymous unless the list only takes reservations from signed-in people, and the owner sees what's reserved when they choose to look.
If your name will be on it, the confirm window tells you before you reserve: "This reservation carries your name." If the owner is already looking, it says "The owner can see what's reserved on this list, so they'll see your name." There's more on how gift reservations stay a surprise.
How can I tell if the owner is looking?
The wish tells you before you reserve. It says either "[name] can't see what's reserved" or "[name] can see what's reserved on this list, so reserving this won't be a surprise." Not signed in on a list that asks you to sign in? Then it reads "Reservations on this list carry a name" or "[name] can see what's reserved on this list, so they'll see who reserved this." While they're looking, the top of the list also says "[name] sees what's reserved".
After you reserve, the wish keeps you posted: "You reserved this. [name] still can't see it." or "You reserved this. [name] can see it's reserved." Open it again any time to check.
Can I leave a note for the owner?
Yes, but only right after reserving, in the Reserved window:
Reserve the wish (see "How do I reserve a gift?").
In the Reserved window, tap Leave a message for the owner.
Type your note and tap Leave message. You'll see "Message sent", with Edit if you want to change it.
Close the window when you're happy with it. After that, you can't add or change the note, from the wish or anywhere else.
The window also says when they'll see it: with your reservation if they're already looking, or whenever they choose to look. They can't reply, since Wishpicks has no chat between owners and gifters.
Can other gifters see it was me?
No. They see the wish as Reserved, with a faded photo, no name and no note, while you see it as Reserved by you. When they open a wish someone else reserved, it says "Someone else reserved this." with a "Find something else for [name]" button that shows what's still free.
Tips
On a list that doesn't ask you to sign in, sign your note if you want the owner to know who it's from. Nothing else on the reservation names you.
Write the note so it's fine to read early: the owner may see it as soon as they choose to look.
Need their address to send the gift? See "Does Wishpicks deliver the gift?".