No. Wishpicks doesn't ship gifts and never handles payment. The gifter buys the item at the store, pays for it and any shipping there, then works out delivery with you.
Wishpicks helps people choose the right gift. It isn't a shop, so there's no cart and no checkout for gifts here.
How a gift gets to you
Someone opens your shared wishlist and reserves an item. That only marks that they plan to get it.
They click through to the store and buy it there. Whatever the store charges for the item and for shipping, the gifter pays at the store's checkout.
They arrange delivery with you. They can ship it, bring it in person, or ask where to send it.
What about Wishpicks Plus?
Plus is the one paid thing on Wishpicks, and it's only about price tracking. It doesn't buy, ship, or deliver anything, and it has no bearing on what shipping costs.
Good to know
Reserving isn't buying, and it doesn't fulfill the wish either. Wishpicks can't confirm that a purchase actually happened at the store, so a Reserved badge means "someone plans to get this," not "this has been bought." You mark an item as received yourself, once it turns up.
Shipping cost, speed, and where it ships from are all down to the store the gifter buys from.
Related
"How will someone know where to send my gift?"
"How to safely share delivery details"