Sending data over two independently powered arduinos I have multiple arduinos that are each powered by a 12v battery through a 5v regulator.  All arduinos share a common ground.  Can I simply connect the digital output from one arduino into the digital input of another?  I'm worried the (small) voltage differences between each regulator could cause a problem.
It's ok if the distance between the two devices are not too big. (couple meters)

Since the valid logic level (voltage) ranges are pretty wide a small difference won't affect the operation of the devices. Arduino uses TTL (5V) logic so let's see the level diagrams:



If you see the output level ranges (on the right) are more strict than on the input side (on left) so the logic value remains valid even if a small voltage drop or noise occurs on the wire.

Make sure you won't connect two outputs directly since it can damage the microcontrollers. Connecting two inputs are totally safe.

EDIT: If you want to cover longer distances you need to use a balanced line  e.g. differential I/O-s, RS-485 or CAN. These are not default Arduino features so you'll need additional electronic to use them.