Two motors in one set of pins/plugs
The drive capacity of the Motor Shield should match the motors – two of the motors. The shield was probably designed for a load of one motor, and you're giving two, so it is possible that neither of the motors on that side is getting enough current to get started. Here are somethings to check to get started:


Find the drive capacity of the Motor Shield and compare it to the motors (if you don't have specs for the motors you could measure the current they draw if you have an DMM). It might be wise to measure the current anyway, just in case you have a defective motor or shield.
If the capacity or the current draw is out of line, that's your problem. You can probably just add a power transistor to the circuit to drive the motors in this case.
Try it with just one motor on each output and see if they run normally. If they do, that would suggest a loading problem or a defective motor or shield. You can sort out which is which by swapping sides – if one motor runs and the other doesn't, swap them. If the problem follows the motor, then blame the motor, otherwise blame the shield.