Choose them for the shipping address and buy it! How do you secretly buy something on Amazon? Solution is to remove the delivery address from your list.
Amazon wish list is a gift archive, just like a birthday or wedding gift catalog where you create a list of all the things you wish you have, and when friends and family want to buy a gift for you, they can quickly go through your wish list to select from what you listed there.
Tap Wish List or whichever list you want to share.
You will also need to know your destination country and a range of shipping date within 6 months.