Effective communication is a necessity. An english class is a dumb measure of being able to use the language (in my opinion). If you can comprehend (easily and consistently) what you are being told, and if you can explicitly communicate what you desire to convey, then I wouldn't see there being an issue. I think the only way to discover this information is to hold an interview with the person.
However, yes, if you are unable to do the above, I think it should preclude employment.
That said, there are several asians and europeans that work as either post-docs or grad students at my university, and communicating with them in english isn't always the easiest. However, in my lab I mostly work with germans, and my PI speaks fluent german, so there is another method of communication for them. I'm not sure if this is the case in the other labs.
If you're talking about yourself, you seem to have a near mastery of the language, at least on paper. Is a thick accent what you're concerned about?