[Al Skierkiewicz]

OK,
First things first. You first picture clearly shows a wire going to the gearbox and it looks strangely like the red lead of a Chalupa motor. This is a big no no and you will not be able to compete with this arrangement. Next you have a screw missing from motor(+) on one of the controllers which leads me to believe you using a common ground or are wiring motors in series. This is also not allowed and will not work if you plan on backing up. The motor leads on the controllers change polarity, i.e. sometimes the motor(-) is battery negative and sometimes it is battery positive. Lastly, as someone has pointed out, the mounting hardware on your controllers is touching motor(-) again giving a sneak path to frame. 4-40 allen screws work OK but tywraps work and are non conductive.
So what was happening? If all is as I surmise above, the motors on the left side are reversed so that the robot can drive forward. With the polarity reversed and the motor and controllers touching the frame, the output of the left side controllers are shorted when driving forward. The opposite condition would be the case if you were driving backward. Make the suggested fixes and try again. That is how the LEDs on the disconnected controllers were lighting when you ran the right side.
As to the question about magnetic fields...the answer is no. Yes a wire in the presence of a magnetic field may have current induced in the wire. This is dependent on two things, the magnitude of the field and the field must be changing or the wire moving. Since the motor is a permanent magnet, the field is not changing, and since the wire is not moving there can be no current induced in the wire. Since the motor leads are connected to a fairly low impedance, any induced current would be tiny compared to the motor run current.
If you have any further questions you can always PM me. I won't have an immediate answer but I will get back to you. Anyone can ask me questions and I will try to answer.