Can't believe this thread is still going.
"Ground" is just a fancy term for "reference". Seems like this discussion comes up every once in a while. Here's an experiment that sometimes helps.
You have a 12V battery and a DC voltmeter. You want to know the voltage potential at the + terminal. So you take the red lead of the meter and touch it to the + terminal. You leave the black lead touching nothing, because "you don't care about the - terminal, just the +". What does the meter read? Zero. Why? Because voltage measures the DIFFERENCE between two points. A measurement - almost by definition - has to be relative to something, some agreed-upon baseline reference point. When you stick the positive lead on the + terminal, the - lead is connected to nothing... and there is no voltage between the + terminal and nothing, so the meter correctly reads zero.
Now, you touch the black lead to the - terminal of the battery and magically, you see 12VDC. Why? Because now the meter sees a difference in voltage between its two leads.
Next you touch the black lead to the + terminal along with the red lead. They're both touching the same terminal. The meter reads... zero, because there's no DIFFERENCE between the two leads, they're both touching the same thing.
Now let's grab a second 12V battery. And let's connect Battery A's - terminal to Battery B's + terminal. Now you effectively have a three-terminal battery: A + terminal at one end, a "middle" terminal, and a - terminal at the other end. If put both of the leads on the + terminal, what will the meter display? Zero, because there is no DIFFERENCE between the leads when they're touching a shared conductive object. If put both of the leads on the - terminal, what will the meter display? Zero again, because there is no DIFFERENCE between the leads when they're touching a shared conductive object.
Now put the black lead on the - terminal and the red lead on the "middle" terminal. The meter reads 12V... even though there's a whole second battery right there! Why? Because the DIFFERENCE the leads see is just the difference between the terminals of one battery. Since the other battery is not "in the circuit", it's as if it doesn't exist to the meter. You can connect it, and disconnect it, and wave it around, and carry it outside, and the meter won't know or care.
Put the read lead on the + terminal while leaving the black lead on the - terminal. The meter reads 24V. Why? Because you're measuring across BOTH batteries. The first battery has a DIFFERENCE of 12V between its terminals, the second battery also has a DIFFERENCE of 12V between its terminals, and since you've effectively "stacked" the two batteries, the meter "sees" both differences and sums them.
Now we come to the heart of the matter: Move the black lead from the - terminal to the "middle" terminal. What the heck? The meter now reads 12V! But the red lead is still on the + terminal, the one that was giving us 24V just a moment ago!?! How can the same + terminal yield two voltages? The answer is, because the + terminal -
by itself - is meaningless. The + terminal only has a "voltage" RELATIVE TO SOMETHING ELSE. The meter reads the DIFFERENCE between two points, remember? So yes, you left the red lead on the + terminal but you moved the black lead to a different place on the battery... and so the difference between that new place and the + terminal is not the same voltage as the difference between the two places you were measuring just a moment ago.
We can call the - terminal "ground", and then we have two different voltages we can measure that are both positive relative to our agreed-upon ground point. Or, we can agree to call the "middle" terminal "ground" and then we have two different voltages, one of which is positive relative to "ground" (the + terminal) and one of which is negative relative to "ground" (the - terminal). Heck, we could even agree to call the + terminal "ground" and then we'd have two different voltages that are both negative relative to "ground".
If you've followed along here, you should now understand that voltage is a measurement of the DIFFERENCE in potential between two points. We can arbitrarily refer to some voltage as "ground" and then measure everything relative to that point.
Hopefully now you recognize that a single contact between two systems won't convey any information. It's just like the very first step above, touching just the red lead to the + terminal... the meter read zero because it could not discern any DIFFERENCE between its two leads. That + terminal might be at 1V, 12V, 1000V, could be AC or DC, could be anything, but the meter cannot SEE any of that until it can compare its red lead to its black lead.
Likewise with your two Arduinos. Connecting a single wire between two otherwise isolated systems doesn't give them any way to measure DIFFERENCE on that wire. They need a shared reference point, an agreed-upon baseline.
To avoid confusion: There are many ways - some intentional, others not!
- to provide this reference point. Connecting a second "data" wire might, in fact, work under some circumstances because now you've established a two-point connection and under a magic set of conditions each might provide a reference (really, a current path) for the other. But it's unwise to rely on "magic" in electronics. Hopefully, someone's earlier and excellent example of a balanced differential pair (without a formal ground) will now also make sense because the receiving system does, in fact, have TWO points and can measure the difference between them.
Bottom line: You really do need to provide a common reference. Ground is almost always that common reference except in special cases like differential pairs. For your Arduinos, it's best to connect their grounds - with the caveat that you need to watch out for ground loops and the like, which is an entirely separate topic.
Hope this helps.