@Duke the OVRdrop program allows you to display a window like others mentioned above that only you can see and looks like one of the windows in elite dangerous that opens when you look at it , otherwise it is invisible or transparent , and tiny so it doesn't obstruct your game. Nobody else can see this. you can pop out chat in your own stream then close the main stream window and just capture that chat box so you are not wasting bandwidth watching your own stream.
dunno if that application works in the Rift but others on here who use the rift can let you know!
as for capturing your game to stream, any game you play grabs a 2 d mirrored version of itself that is displayed on your monitor and THIS is what OBS captures and puts out to stream.
an example below of what iracing looks like for example:
I recorded this to show a friend what VR and iRacing looks like but it should show what I mean just for the purpose of showing what OBS puts out.