Yeah, use a scene with a Game Capture for the in-game stuff, a second scene with a Window Capture for the PVP.net client and Twitch chat, and set up a border/background graphic. Will look more professional, and less lazy and hackneyed (even if a number of large League streamers go in for the lazy-ass method, it looks like crap).
Also, look into the Automatic Scene Switcher plugin so it'll auto-flip between your in-game and out-of-game scenes as you load into a game and after you exit out on win/loss.