I'm seeing some confusion here. You do not need two PCs to play games using SteamOS.
You know how some games support MacOS, and some games only support Windows? SteamOS will be exactly the same. Some games will support SteamOS natively. Some games will not.
If your game does support SteamOS, you can install it directly to the device, just as you do with Steam now, just as you download games to your gaming consoles, and just as you install games to your smartphones. SteamOS supported games will be purchasable, downloadable, installable, and playable right from the source.
If your game does not support SteamOS, you can install it to a Windows or Mac device as we currently do, and stream the game over a local network to a SteamOS device. So if I had a SteamOS box in my lounge room, but a game that does not support SteamOS, I could turn on my Windows 7 PC in my bedroom, turn on Steam, install the game there, and play it in the lounge room on my SteamOS device by streaming data over the network from my PC.
Though this will obviously be problematic for games that don't support SteamOS, Valve has made a point that they're working closely with developers to ensure games do work with SteamOS. And if a game is native to SteamOS, it can be installed directly to a SteamOS device. Absolutely zero streaming required. Examples, going by the SteamOS home page: Total War: Rome II and Metro: Last Light.