To share or not, grounds of two power supplies
If you are sending a signal via wire from the Arduino to the motor controllers then the current requires a return path to the Arduino. The common ground provides that.

In many applications (particularly industrial) it is common to isolate the power supplies. In that case an opto-isolator is used and the return paths are local to each side of the opto-isolator.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. Opto-isolated connection allows isolated power-supply grounds.

The risk with common grounding in your case is that under some fault condition
large motor return currents use your micro as the return path to the motor power supply. This could, for example, burn the relatively light PCB traces on your micro or introduce significant noise on the ground lines and interfere with operation. Good wiring layout can minimise the risk of this.



The concept of separating power and logic power supplies is very simple.







simulate this circuit

Figure 2. Bad wiring practice. All the motor current returns through the micro-controller board risking interference and overheating of low-current traces.







simulate this circuit

Figure 3. Good wiring practice. Here all the motor current only flows in the 12 V loop. The micro and motor control board power-supply grounds are connected at one point only (shown in green) so that the control signals have a return path.