Recently I was playing with an ESP8266-01 module, and could not communicate with it over serial using an FT232RL-M board @3.3V. These were the symptoms:
It booted, I could read the serial output on 115200 baud.
If I sent any AT command, it just echoed the command back.
My first setup did not work because the power supplied from the FTDI board was not enough. So my second attempt I added a 3.3V power supply to the setup and ran the ESP board from it. Now everything was working, except the serial commands. After changing the wiring again I created common 3.3V and GND to sync the voltage levels. This way the problem was solved.
I have seen many people running into this problem over the internet, but with no clear solutions.