Well, it depends on the country. In Australia for example you definitely need a degree to be able to teach, both in public and private schools.
Anyway, CELTA is the preferred qualification in teaching English as a second language by employers, so you can be sure that after having taken it you will be able to find a job. I suggest you have a look at this website to get more info about TEFL courses around the world and job opportunities after CELTA:
CELTA Teacher Training Courses, Cambridge DELTA Diploma, Cambridge ESOL, English Language Teaching to Adults, Tesol Courses, Tesol Certificate, English for Tesol, Cambridge TKT test, TEFL Certificate, TEFL Course, Sydney, Brisbane, Australia, Seville