Drivers, GeForce Experience and Steam are all up to date.
Whenever I do PC game streaming, my 360 controller and the default SHIELD controller work just fine while in Steam Big Picture Mode only. Once any game is launched via Steam the controller input stops working completely. For both my 360 controller and the SHIELD controller. Even if the controller is plugged into the PC itself (while streaming), no controller input. Yes, the games have controller support; I've played them via controller before while streaming. Controller works fine in games when not streaming (playing locally on PC). Changing from Steam stable release to Steam beta release didn't do anything. This is a recent issue, as I have streamed via the SHIELD before playing with my controllers without any problems. I suspect this just came into effect with the latest Steam update. I do not know if the issue exists on Nvidia's side or Valve's.
2 games I was trying to play are Tomb Raider (2013) and Resident Evil Revelations 1
If the same is happening to you, please report it here:
Update:
OK I figured it out, but this led to the discovery of a new problem.
Once the stream was connected and I was in Big Picture mode, I had to go into mouse mode with the SHIELD controller and turn off 'Multiple Controllers' support. Now once I launch a game, both controllers continue to function.
However - the 360 controller doesn't work properly in games. The key bindings are off/out of whack. And for example - in Resident Evil Revelations 1 - when using the 360 controller - without hitting any input - the camera will just spin around continuously. When I start using the SHIELD controller - it stops and works properly (the SHIELD controller works and the camera stops spinning). Also, the 360 Dpad does not function in any game it seems.
Using the Moonlight Game Streaming app on the SHIELD ( https://play.google.com/store/apps/details?id=com.limelight&hl=en ) - the wireless 360 controller works 100% properly in all tested games.
So, it appears the 360 controller does not function properly only when streaming through GeForce PC Games. Hopefully they fix this.
I do not have any issues with using my controllers emilee streaming via Steam. I haven't tried paying with the games you have issues with though, but I've played many with no issues using the Shield controller.