AIUI there are at least four possible "chip" transaction types, chip only, offline chip and pin (card verifies the pin), online chip and pin (bank servers verify the pin) and chip and signature. Some cards and some terminals only support a subset.
The US has mostly gone for chip and signature as the default, Europe has mostly gone for chip and pin as the default (not sure if it's online or offline). I have heard (I can't confirm because I'm not an american) that with some banks even if you have a pin it only works for ATM/EFTPOS transactions not for EMV chip and pin transactions.
So if you bring an American card to Europe then you can have problems, if it's a manned terminal it will probablly do C+S which the terminal will handle fine, but the operator of the terminal may find confusing. If it's an unmanned terminal, then it might process the transaction as chip only or it might refuse it completely.