Ok.... I'm now thoroughly confused.... I thought the ESP8266 was the chip ON the WiFi sheild?
So I'm now thinking I need to use the ESP8266 library to turn the 8266 module into an AP, then can use the WiFi and WiFiUDP libraries on the Mega to send and receive packets.Is that even close to correct?
the "server" is using one of these, and no AVR Arduino.
So, it appears I can implement the entire functionality using the ESP8266 library. Though it's not obvious to me how to do UDP on this, as there seems to be no UDP class as part of the ESP8266 library.