The current will be consumed by the LEDs, not routed back through ground.
The way to guarantee this is to be sure the - pin ["ground"] of the separate power supply is connected
to the LCD grounds and not to the Arduino board ground.
Also, with this system, you run the risk of creating a big ground loop that produces oscillations and noise
in the system. The way to prevent this is to use a Y or octopus arrangement [visualize it], where you have
a "single common ground point", and you run a power and ground wire to each of the subsystems separately,
including the Arduino board. The worst arrangement would be to daisy-chain the power+ground from one
board to the next, and then loop the last one back around to the starting point. Now, you have big ground