Gon and Kurapika's examiner also taught them nen. Is every examiner also a teacher, or the hunters have to learn nen by themselves?
The examiners aren't there to teach the hunter candidates anything, they are meant to observe those hunters who haven't yet gained nen for a limited period (I forgot the duration, but its a few years iirc) to see if they succeed in progressing and then give them the actual Hunter License.
I think Kurapika learns it from someone other than his examiner. Also of course there are others like Hisoka who already know it and are qualified (provided they pass the earlier exam) as hunters already.
That said, I don't think the hunters have to learn nen by themselves, they just have to understand its existence and then search for teachers. I do believe that all examiners have the capability to teach the candidates. Whether this is also an obligation if they're approached for guidance, I'm not sure ( I think not, need to re-read Wing & Gon's interactions to recheck )
I belive you're refering to the "Secret Exam" when you talk about "examiners". I'm not sure this is entirely accurate.
There doesn't seem to be a very strict structure to the Secret Exam, at least compared to the Public Exam. As such, there doesn't really seem to be "examiners" as such. The hunters just seem to collectively keep tabs on the passing participants to see if they figure things out on their own. Some actually teach these people Nen, but none seem exactly obligated to do so.
So, in short, there propably isn't the sort of requirement you mention, because there aren't really any "examiners" at this point. All teachers we see seem to be giving their lessions more or less out of their free will, rather than any sort of obligation. Of coures, almost everybody capable of teaching Nen is already a hunter (with the some exceptions like the Phantom Troupe, who know Nen but aren't necessarily hunters), so they would be qualified to pass those they teach, when they seem ready. But remember that there are people who already know Nen, who pass the Secret Exam the moment they pass the Public Exam, without any sort of fanfare.
Basicly and simplified to the extreme, I'm proposing that in stead of any sort of examiners, just about any hunter can just contact the Hunter Association and say "Yeah, they know Nen now. They're real hunters alright." Of course, the exact details of the Secret Exam are rather vague, so the answer is rather speculative.
As a side note, I think it would be theoretically possible for someone to pass the Public Exam, learn Nen from a non-hunter and then pass the Secret Exam when other hunters learn about this. But there isn't really an example of such thing ever happening.