RayLivingston:
Ok.... I'm now thoroughly confused.... I thought the ESP8266 was the chip ON the WiFi sheild?
There are ESP8266 shields you can buy but they are not official Arduino products. The official Arduino WiFi Shield uses the HDG204 so it's completely different and that's the shield the WiFi library included with the Arduino IDE is compatible with.
RayLivingston:
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?
You can certainly use the ESP8266 core for Arduino and the ESP8266WiFi library to write a firmware sketch and upload it to the ESP8266 modules if you like and then use whatever communications protocol you have devised to communicate between your AVR Arduino boards and the ESP8266 modules. The more common alternative is to just use the standard AT firmware which most ESP8266 modules come already flashed with. The WiFiEsp library makes it very easy to control ESP8266 modules running that firmware by hiding the AT commands behind the standard API used in the official WiFi and Ethernet libraries. You can see an example of that library being used with an AP on the ESP8266 module in this example sketch:
But you need to get this idea of using the standard WiFi library out of your head. That said, because of the similar API it is possible to reuse code written for those libraries with the WiFiEsp or ESP8266WiFi libraries with minimal changes.