I usually open 2 windows and have them both on screen side-by-side. But there is a delay on the livestream from the game so it can be confusing.
If you want to participate in the party then I'd say stay in the game and watch the livestream later.
yes otherwise you have both sounds through eachother. Ofcourse if you dont mind that you can keep the game unmuted but personally I find it very annoying to have 2 entirely different sounds playing through eachother, especially when one is talking