No, not likely. The "native" speaker thing is just a "nitch". If they want non-native speakers, it is most likely they will just hire a citizen of their own country. Many people in Europe know English very well. SO why would they want to hire someone from Italy, when they can get someone with the same qualifications and can speak the mother language of the students.

For example, if you want to teach in Germany, why would they hire an Italian, when they could easily hire a German who has the same qualifications in English teaching as you?

You could most likly get a job in Europe in your own country.