Some 11 years ago, when I had to get the FAA A&P licence, the biggest problem for me as a European was not the technical knowledge (I already had an Irish AME licence and a JAR 66 B1), but to get a letter from an American operator to the FAA, which stated that I, as a foreigner, was required to maintain their aircraft and that no American national was available to do the job.
Mind, at this time I was working for a big American cargo airline on their European base, a fact which was at least known to the FAA field representative, who had to issue the certificate to allow me to take the exams.
I donīt think that the EASA put any of such obstacles in the way ( it is more likely up to the individual EU countries to regulate the access of non-EU citizens to their markets through labour laws, not through aviation laws).
Similarly I had American colleagues, who lived in Europe and were bitching about the fact that they had to get an EASA licence (or JAR licence at this time) to get a wellpaying job certifying for European aircraft (there are not too many maintence companies around in Europe where one can get a good job solely based on a FAA A&P licence).
At least they didnīt face obstacles like having to prove their need to get the European licence, they were just required to get their behinds into gear and take the exams.