First, the reason not to have power on the Arduino microcontroller pins when power is off is due to phantom-power which can cause all manner of bizarre happenings!
"Phantom Power" to Arduino via input pin??? - Using Arduino / Microcontrollers - Arduino Forum
Connecting 2 Arduinos via serial (Rx-->TX, Tx-->Rx) is often done. Beware EMF interference from AC florescent, motors, industrial equipment... Even lightning! Remember, long unshielded wiring is an antenna just waiting for a signal.
Now, for serial-serial, you can avoid phantom-power issues by using an opto-isolator to isolate the serial line . Optos will however limit the BAUD rate, testing required.
Caveat:
The above cut 'n paste graphic from a stackexchange post should not be taken verbatim with respect to external resistor values. After a back-n-forth discussion with Paul_B, I strongly suggest that the datasheet for the opto-issolator selected be consulted and the collector load resistor be chosen based on the information presented.