Few things to check out.
1st, when you talk in league voice does the green ring appear around your icon? If not, check your mic and try turning off then back on your headset with the mic off, then turn the mic on after it finishes connecting.
2nd, did your friend check their sound settings to make sure the league client is turned up?
3rd, your friend should make sure they can hear other sounds through the headset.
4th, if all of the others are correct.... have them open league AFTER turning on headset. There have been some people whos headset bugs on client because the headset was turned on after league is up. I have that issue with another game, could be a windows update issue? Who knows atm.