Yes, you just need to swipe left on the person’s name on the Sharing screen.
This applies to any kind of list e.g. wish list, shopping list, etc.

There are various ways of adding a wish. Tap + bottom right and fill in the form, type in or paste a URL at the top of the screen, or scan a bar code of a product.

You can swipe right to mark it as bought, so that other people don’t buy the same wish.

Yes, you can either send an email with the content of the list or an invitation code to get the app.

You can share your wish list in the menu by tapping Share list. Then you can type in the user name or e-mail address of the person.