If I understand you right then you want the circuit to come ON if either of sources want it to be on.
SO for basics I would use relays in paralel with its relevant transistor and diodes as protection. 
Signal from controler is connected to the base with suitable resistor (for arduinos mine favourite is usually 1K) 
The relays have solenoids + connected to the supply and ground to collector.
Emitor is grounded.
the diode goes in OPOSITE direction between relays winding to protect the transistor when it switches off

 will be in parallel and As on "OR" System: if either is final circuit is on. each can only switch its own relay so can only switch its own relay off. All three circuits are fully isolated. 

Sorry for no schema, the tool does not work for me