Privacy works by purchasing blocks of prepaid cards and issuing them on demand. So while your bank is linked to the card, the issues cards are not linked to your identity. With privacy, you can use any name and address you like and it will be validated. They use the card number, the CVV and the expiration as the only means of validation.

So this means I can say my name is Fake Customer and I live at 404 Main St, 95111 and it will validate just fine.

This is the same as if you purchased a prepaid visa gift card.