(New EU: Bulgaria, the Czech Republic, Estonia, Hungary, Latvia, Lithuania, Poland, Romania, Slovakia and Slovenia; Pre-EU: Belarus, Bosnia Herzegovina, Croatia, Macedonia, Moldova, Montenegro, Russia, Serbia and Ukraine)
Also see TEFL In Europe and TEFL In Western Europe; for specifics on Eastern Europe read on here.
There is still seen to be a need for English teachers in the new EU states; while it is substantially easier for schools to employ EU teachers, it is generally still possible in these countries for non-EU teachers to get permission to work. But get your qualifications, documents and certificate of no criminal record ready – and apply well in advance because there is some monster bureaucracy in store for you!
In the pre-EU states, things vary. In theory the procedure should be the same for any teacher seeking permission to work, and this means nationals of one country will not have any clear advantage over others. In countries with less of a history of native speaker teachers, there may be fewer jobs, and native speaker teachers may find themselves expected to do the same as a local teacher, including teach grammar. However each country will have its own regulations – regulations which may have been developed with other priorities than facilitating the work permits of native English-speaking EFL teachers. It’s not the EU and in general this is a positive thing for non-EU teachers, but you need to examine each country individually.
Salaries vary quite a bit, making generalization difficult, as usual. Overall you are more likely to get a full-time position, with a flat included, in Eastern Europe than Western, but nowadays you can’t really count on this in Eastern Europe either. It’s unlikely that you will be able to save much in any position in Europe – and even if you manage to save a substantial portion in Eastern Europe it may not amount to much at home.