"Cool" idea

Make it so I can hear my own team mates through the screen when I’m in private chat instead of just the other team

this is already possible, guide - preferences - voice - play through headset and speakers

Haunted, that is general playthrough. Halfway is saying be in party with friends, then hear only his teammates through the TV. I am surprised it doesn’t work this way, but if I am not mistaken there are options to who you can hear when. Enemies are only heard in social playlists (all but Arena probably for Halo Reach, but social in Halo 3) with close proximity. Basically when you put the game audio through the TV you currently, you get your team + enemy proximity. I don’t think this will change, but there could always be an option to always ignore the other team.

For now, to get the same effect, simply mute the other team. Frustrating, yes, but then you will only hear your team and not others in close proximity.