Let me try to help you.
Why do people use Discord?
It has become the default application for voice chat while gaming on PC, like TeamSpeak and Mumble were a few years back.
Destiny's in-game voice chat runs through peer-to-peer and relies on every player having a properly configured network and their internet service provider providing them with a decent NAT service when a public ipv4 or a ipv6 address are not available. Discord has servers instead, so it's simpler for everyone.
Why you can't hear anyone?
Discord picks up your default audio device in Windows for output by default. Destiny picks up your default audio device for the game sounds and the default comms device for voice chat. If you set up in Windows' Audio Control Panel your headset, for example, as the default audio device and default comms device, it should work without issues. If you still have issues, in Discord's settings you can choose your output and input devices too. The "Voice & Video" tab will allow you to select from a drop-down menu your output and input devices, control volumes and some other features. There's also a mic test, you can use it to test how people will hear you and if it's outputting to the correct audio device.
I see that you have a clan, my recomendation is that you setup a Discord server for your clan, it's free, and there you can get more familiar with how it works. If you have any questions, feel free to ask me.
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Thank you for taking the time to respond. I was successful yesterday in getting discord to work. I really didn't have to do anything special. But I did notice that I was able to be heard through both Discord and in game. Someone said they knew when I spoke because their was an echo. Peer to peer vs servers make complete sence. Thanks again.
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Try enabling push-to-talk in Destiny, you set up a key to press when you want to use the game's voice chat.