The TCP solution posted relating to communication between two ESP8266 seems ideal for me given the fruitless and substantial time I too have wasted attempting TCP Client-Server communications using ESP AT commands.
However, I am getting compilation errors when attempting to upload the Arduino sketches (modified) to my Arduino Nano's: 'Error compiling for board Arduino Nano'. So is the ESP8266_TCP library hardware specific? Did you have success with Arduino Uno boards and so that is my problem. I have downloaded the ESP8266_TCP library Zip files and have installed them successfully using the Arduino Boards Manager. I have an Arduino-ESP8266-01 configured as a server and likewise for the TCP client.
Hope you can shed some light because this is the most promising solution yet and I would like to focus on getting it to work.
Thanks in anticipation of suggestions.