Hi there,
the problems of teaching EFL as a non-native English speaker start when you want to teach outside Europe. I am German myself and I found that many countries only give work-visas to EFL teachers with a British, USA or Australian passport.
The other problem is that as a non-native your CV will probably always end on the 'second' pile - understandably so. People who come to Ireland to learn English expect an Irish teacher, not an Italian. This is not to say that the Italian teacher isn't as good as an Irish teacher, but you'll have a harder job convincing them of your qualities.
This is why it's essential that you have high-level qualifications, ideally many years of solid teaching experience, AND - speak the language as fluently as an educated native speaker!
Wish us luck!