People who do not speak English as a native language tend to have an accent, but some may be so fluent in English that they manage to speak in a certain accent like British or General American. Some may choose to speak with an "approved" accent, others might keep their native one.
I feel like I am being fake if I choose to speak with a British accent (which would be the most natural accent for me to adopt since I've been taught BE in school). I know how to do it, but I choose to speak with my native accent because that seems to me more real.
I find it charming when others speak perfect English, but with their own accent. An example is Christoph Walz, which is very fluent in English, but has his distinct Austrian accent. I'm sure you have seen him in Inglorious Basterds and Django Unchained, but here is an interview with him.
So my question to you, especially to you who are native speakers of English, what do you think of people keeping their non-English accent? Should they do it, or should they adopt a native English accent?