I have had this issue multiple times and I really want to know what is going on. So I have a bench power supply, and I was powering my ESP8266 off of the bench supply. I then wanted to send it some AT commands, so I set up the Software Serial on the arduino and hooked everything up. All i got on the serial was complete garbage. I decide to disconnect the ESP8266 from the bench supply, and use the 3.3 output of the Arduino. I reset everything, and BOOM, serial monitor is working perfectly and I am no longer getting garbage from the ESP.
This has happened with multiple modules and I am at a total loss. Why would everything work fine when the module is running off of the 3.3v output of the arduino, and NOT when powered from 3.3v off of the bench power supply? Thank you for any clarification!