on my credit cards (including Costco) I have alerts set up that text me when a purchase is made over a specific amount or made without the card present (eg. just giving the numbers and expiration date).
My Costco card has only been used at Costco, I have a mileage rewards card I use everywhere else. A few months ago I got a text one afternoon saying I had just made a purchase w/o the card present in Morocco! I immediately called Citi, they cancelled the card and sent out another by Fedex. Have no idea how they could have got my card number…