My friends and I were able to get the Xbox Live party chat working during the Destiny beta. As long as the party was left snapped to the side of the screen, the voice chat worked well, even with the Destiny beta running.
Yeah its a common problem. I think what needs to be done is that everyone needs to change from party chat to game chat. I found that even that ended with issues though. To avoid the problem stop using the party chat system with people who are playing the same game as you. In Destiny just form a Fireteam with those people and it should work fine. Only use party chat if people are doing other things, i.e. Using Netflix or playing another game.
Edited by Canadian iPoo: 7/31/2014 3:14:01 AM