Hi,
I just now had this problem, and I found a solution. After downloading google speech recognition and synthesis , and enabling microphone permissions on google play services, Duolingo speech exercises still weren't working. So I opened my browser, and pressed the microphone button on the address bar, and a pop-up asking for permissions for google speech to use the microphone came up. After saying "only while using the app", I went back to duolingo, and it works fine! I hope this helps.