I have read so much information on this site about debit and credit cards, that I had to write this. Bear with me.
First, the European cards have a chip in them that is not available in the US. That means that a US card cannot be used in self serve purchase locations, like a gas pump. But US cards work everywhere else--ATMs, purchases with a person, etc.
If you do not live in a specific country, you cannot apply for credit there, as there is no credit history on which to base a credit decision. Therefore, a US citizen has very little chance of obtaining a European card with a chip.
Next, your cards are monitored by a computer system that looks at each purchase to determine if it is legitimate. It looks at location, amount, even as detailed as the store. If the purchase or location does not match YOUR patterns, the system will pick it out and an agent will call to verify the transaction. If the agent cannot reach you, it will protect your account and you will not be able to use the card--it will be frozen. That is why it is absolutely vital that you alert your debit and credit card issuers before you leave--even for another US state. (At least they can call you in the US, if they have a cell number!) If you don't call, your cards will be blocked and useless.