When I'm playing a video game, I want to be immersed in it. It's a very clever illusion, though one I doubt anyone ever consciously invented, that since your field of view is always in the same place in retina-space[0], a static screen can trick you into thinking you're looking or moving around by displaying moving images.
I feel like having a second screen, especially one I have to hold in my hands, could only distract me from this effect and remind me that I'm actually sitting on a couch, staring at a big lighted box... which doesn't really sound like fun to me.
[0] Seriously, think about this. It blew my mind.