You can't do it directly between two cards, at least at the level where all you have is the card numbers. That company does it by sending the recipient an ATM card that they created themselves, and when you give them money they can put that directly into the bank account behind the ATM card.
In a sense, you'll have to do it the same way. Some banks offer an external transfer feature as part of their online account access. If you own the destination card's account and know its direct deposit number, you may be able to push money to that card from your checking account. (But don't assume it'll show up instantly.)