Current flow between 2 circuits without common ground
That video is 10 minutes long so not many will watch it for you. A screengrab of the relevant part of the circuit would be good and a sharper photo. 

Anyway, congratulations for realising that there should be a problem with the ground line removed. You may have enough capacitance between the two circuits to hold the grounds steady enough relative to each other to let it work - particularly if they're both close to a conductive surface. Try separating them or changing orientation and see if it still works.

Report back in your question.




  How far do you think, should they be?


I would expect that capacitance would be very small at quite short distances - 10 cm, say. There will be a little capacitance to everything in its surroundings - including you. The input of your micro-controller is very high impedance (resistance) so it will be very sensitive to stray voltages. If you add a resistor from input to ground I think you will find that there isn't enough power in the stray voltages to turn on the input.