Although it's not impossible, it is much harder to find work in Europe if you don't have a passport from an EU member state.
Most schools will simply not have the time nor inclination to go through the hoops to sort out the visa issue. Another problem is that in some cases your visa will be tied to the school so, for example, you are not able to move jobs and stay in the country.
Having said that there are a number of non-European passport holders working in schools. Most, from my experience, have come over and found work by knocking on doors rather than through an online job advert (and many have at least a girl/boyfriend or spouse in the country where they're working).
The idea here is that they have made themselves indispensable to the school; first by turning up on the doorstep when the school is in need and then by providing such a good service that the students want them to stay on. In other words, they're good teachers and give a lot!Working in Europe