Hi '170, Happy New Year!
I've held a JAR licence issued by Portugal for a few years now as you know and got my medical renewal in two other countries: UK and Sweden. It was perfectly fine with my issuing Authority, the Medical examiner simply reports back to his Authority who in turn reports back to your issuing authority. There are indeed discrepancies between countries when it comes to conducting medicals, despite the common standards, but in my experience there have been no objections so far in renewing medicals across Europe. Not sure if this will be affected by EASA. Safe flights!