Is a neutral wire necessary between 2 circuits
There are two ways to try and understand why you need a common reference voltage (what you call "neutral", many call it "ground" or "common").

The Voltage approach:

What is a voltage? It's the difference in potential between two points. The key thing there is the number two. 

If you only have one wire connecting between two circuits it will do nothing at all* because there is no reference for the signal. The ground in one circuit (which your signal is referenced to) has no relationship to the ground in the second circuit. In effect you can't apply a potential difference in the second circuit from the first.

The Current approach:

For a current to flow there must be a complete circuit. Without a second wire there is no complete current path between the two circuits. In essence you have a circuit consisting of a wire (the connection between the two circuits) and an infinite resistance (the second connection which you didn't make). The current through an infinite resistance will, by Ohm's law (\$V=IR\$) be zero regardless of the voltage.