Same issue here, I am currently on:
Windows 7 64bit
Adobe Flash Player
Shockwave Flash 21.0 r0
Mic works correctly on other sites like: onlinemictest.com where I am asked to allow the site to use my mic access and voila, works.
But not Duolingo, it seems that Opera is unable to recognize that Duolingo website is attempting to use mic.
No issue in Chrome.
Is nobody else having the same problem? Or just given up and using other browser for Duolingo?