Electronic – Series resistor on digital signal lines

Two common reasons are signal integrity and current limiting in lazy level conversion.

For signal integrity, any mismatch in impedance of the transmission line formed by a pcb trace and attached components can cause reflections of signal transitions. If these are allowed to bounce back and forth along the trace reflecting off the mismatches at the end for many cycles until they die out, the signals "ring" and may be misinterpreted either by level or as additional edge transitions. Typically an output pin has a lower impedance than the trace and an input pin a higher impedance. If you put a series resistor of value matching the transmission line impedance on the output pin, this will instantaneously form a voltage divider and the voltage of the wavefront traveling down the line will be half the output voltage. At the receiving end, the higher impedance of the input essentially looks like an open circuit, which will produce an in-phase reflection doubling the instantaneous voltage back to the original. But if this reflection is allowed to reach back to the low-impedance output of the driver it would reflect out of phase and constructively interfere, subtracting again and producing ringing. Instead it is absorbed by the series resistor at the driver which is selected to match the line impedance. Such source termination works pretty well in point-to-point connections, but not so well in multipoint ones.

Current limiting in lazy level translation is another common reason. CMOS IC technologies of different generations have different optimal operating voltages, and may have damage limits set by the tiny physical size of the transistors. Additionally, they cannot natively tolerate having an input at a higher voltage than their supply. So most chips are built with tiny diodes from the inputs to the supply to protect against overvoltage. If driving a 3.3v part from a 5v one (or more likely today, driving a 1.2 or 1.8 v one from a 3.3v source) it's tempting to just rely on those diodes to clamp the signal voltage to a safe range. However, they often cannot handle all the current that can potentially be sourced by the higher voltage output, so a series resistor is used to limit the current through the diode.

The input impedance of an Arduino is very high, so there will be very little current flowing through your resistors, so no significant voltage drop. I'd suggest 510 ohms or so to protect the Arduino outputs in case you program both sides as outputs.

You must also connect the grounds of the two arduinos together.