In my work, I have an ESP-8266, Arduino-Uno and a Relay. ESP 8266 and Uno is connected. Relay's data signal is take from arduino's 12th pin. ESP is powered from Uno.
If I use two different power sources for Uno and Relay, Relay does not response for the signal from Uno. (Both have a same common ground)
If relay is powered from Uno, then after load control (power on and off) ESP-8266 goes down.