I think they are ok. Some, as you say, speak better English than the natives, eg Joseph Conrad, Einstein, Nehru... Even some unknowns: Walter Wallich who wrote an amazing English translation of «Simpliccimus» by Grimmelshausen. I have to be honest and say that many speak better English than native English. It would be a priviledge to be taught by them if I were learning English. I think their nationality subterfuge is pardonable.

@Débora. Yes, you are completely right about regional or state English, and if you are after American English or Australian or London or Brazilian or Indian or English English, then pretending would be a deception. There is, however, an educated or academic English, which is nearly the same in all the Englishes. This, to me, is the best English and is the thing to go for.

I believe you have to be honest.

But it does leave some people in a bit of a quandary. I've seen schools advertising for English speakers WHO HOLD A PASSPORT FROM THE USA, UK, AUSTRALIA or NZ, or CANADA. Having such a passport does not really qualify one as a good teacher so I hope they actually check the person speaks English. Some do not.

I'm sure a school would be better off hiring a really good non-native speaker than someone who cannot string a sentence together properly. But often, really good teachers are not even invited for interview because they are not from an English-speaking country.

But one should not misrepresent oneself.

So what's the answer? I have no idea! And I am as guilty as those schools, for when I hire a teacher I always check they are native speakers. I expect my Hindi teacher to be from India, my Brazilian Portuguese teacher to be from Brazil. I have a sneaking suspicion I expect them to be able to trace their Brazilian heritage back several generations. It's weird. The more Brazilian they LOOK also helps. A PALE Brazilian simply wouldn't do it for me!

Send for the psychotherapist. I clearly need one.

Irene,

I wouldn't want anything to do with you either if you accuse me of being dishonest if I somehow don't forewarn you that I have a accent BEFORE we talk.  I wonder what some LEARNERS on this site think.  Somehow your target language will get contaminated if you speak with a non-native?  You're out of your mind.  It's all on you. Accent or not, it's YOU.  Look at yourself for your shortcomings.

If you are a learner, anyone can teach you a thing or two if the person is better than you. That's who you need, someone who is better than you.  That goes for everything. I learned more from an Indian coworker than I did from anyone else. His vocabulary and sentence structures were impressive and beautiful.  Oh yeah, and he was a stutterer.   I spent enough time on this site to know that many "students" here are so damn picky even when they don't have any money to spend.

Some of you are being insulting, so here goes.

I have worked with some smartest people in the world who were Americans, Pakistanis, Indians, Japanese, Syrians, Chinese, etc.. These guys at any one time were responsible for a few million transistors working together in their brain.  I myself have more than one college degree, and APPLY what I've learned to make a living. A very decent living I might add. 

However, I wasn't a child when I moved to the U.S., and thus speak with an accent. According to you, I am not a native, which I really, really don't give rat's a*s about.  Also, according to you, if I ever have a great honor of having you, a NOBODY, as a language partner, I must tell you I am not native. If not, I am being dishonest, and who knows where my dishonesty might lead. What do I think I am going to do? Deceive you into sending me whatever worthless currency you use over there?

As the great John McEnro once said, "You canNOT be serious."