I had what appears to be this problem which resulted in a caller or callee being audible on my end, but they couldn't hear me. the solution was to restart the phone - 6 times a day.
The apple support team were very helpful. We had several calls that lead me through and backup, wipe and restore from iTunes, this did not work. A wipe and install as new phone, then re-add all apps, this worked for a couple of days then went bad again. I have reset my network setting, done a factory reset, no dice. All since 10.0 (and still with 10.1 and now 10.1.1).
Eventually I found the audio call routing was set to automatic and it made me think... is this why it decides to switch off one or both ends of audio for me? (There is usually a tell tale crackle in my wired headset before it stops).
So I have now set this to speaker to see whether this stops the **** thing "automatically" offering no audio from speaker, bluetooth or wired headset.
A total pain.
Would be simpler to fix if it was more reproducible.
Shame really. before iOS 10, iPhone has been a great phone. Now, it's more of an 'i' 😟
JAA