Would you definitely have to be a native speaker in order to be capable of embarking on such a 'career', though?
In Southern Europe, yes afaik. Though I have heard stories of Dutch people and Scandinavians being hired in places like Thailand and other parts of Asia (However, only by 'private institutes' which often are dodgy places where very little 'teaching' goes on. I suspect Opebo knows more about this than I).
Once students have an above basic grasping of the rudiments of English grammar and vocabulary (ie. About 'B1' on the CEFRL
, it isn't too difficult to explain things through English. You can, of course, teach solely through English for basic levels but it is much more difficult (imho). Also, what Belgian said.
The real reason, though, I wouldn't recommend teaching in Italy is that I assume it is like teaching in Spain. And in Spain, the facilities and staff support I got were almost non-existent which lead to, as you can imagine, problems... (Lots of good stories about that, actually).