I can tell you that it will be much easier to find a job without a degree if you are already there. Although I have a degree and my TESOL certification, I was unable to secure a job in Prague from America. I finally just bought my one way ticket and managed to find a job within a week of arriving. I think they are more nervous of you not being serious if they don't get to meet you in person. Its definitely more difficult without a degree, but if you can charm them in person, it will greatly increase your chances. There was one teacher at my school in Prague, Lingua Viva Language Academy, that did not have a degree. So it is possible.
As for an EU passport, I don't necessarily know about Poland, but I can tell you it posed no problem to me in the Czech Republic. And there were tons of jobs in Prague. Have a look at expats.cz for jobs there if you at least want to get close to Poland.