I am rather perplexed at the fixation most educated intellectual people have regarding trying to learn a language from native speakers.
Firstly, native speakers are not necessarily the best teachers of their language. This is because most of them are probably monolingual, which means they have never learnt to speak a second language. Hence they are at a disadvantage to understand how someone can acquire a second language.
Secondly, taking English as an example, I question why you need to learn to speak English from a native speaker. You probably need to speak English with a variety of different people, most of whom are using English as a second language as well. Let me give you this example of a multinational company in China that wanted to enroll their representatives to learn English, insisting on having native English speakers as teachers. I asked them who their staff will be using English with. This was an engineering company, and their technology centers were in Sweden, Italy and Germany. Hence it did not really make sense for them to be taught English by an American, British or Australian. Even when you travel to places like London and New York, Sydney, I am quite sure that more than half the time you will be speaking English to people who are using English as a second language.
Lastly, there is no homogeneity among native English speakers. There is no such thing as a common American accent or dialect because different parts of the USA have different accents. Texas, New York and New Orleans all produce speakers with different accents. Within England itself, there is even a wider spectrum, such as Cockney, Yorkshire, Geordie, and Estuary just to name a few. In Australia and New Zealand, and Canada, the accents are also different. In Zimbabwe, the only medium of education is English, so are they native speakers too?
To further illustrate my point, please watch this video of a young man imitating different native accents: http://www.youtube.com/watch?v=mMc3fOhMDUM