It depends what you mean by successful and what kind of situations you are speaking about. Perhaps the most common situation where native and non-native speakers are comparable is taking a 4 week TEFL course like the CELTA and then looking for language school work after. Non-native speakers often know the grammar jargon etc better than native speakers and are sometimes better at grading their language but have sometimes been taught old-fashioned English or in an old-fashioned way and try to pass that on, but none of those things are always true. Some non-native speakers find struggling with the language and the teaching at the same time just too much, so Cambridge Proficiency really is the minimum level to enter the course.
Native speakers often have a better feel for what language sounds natural and are more of a "blank slate" willing to accept ways of learning a language presented in the course, but again it really depends on the person.
The next stage is getting a job. There it is much more difficult, but not impossible, for non-native speakers (just as it is often really difficult, but not impossible, for foreign native speakers to get permanent state school teaching positions).