Not seeing frames returned from client when using ESP8266 Wifi Shield
So I figured out the answer to my own question.

It turns out there must be a mistake in the example sketches that come with the ESP8266 shield I bought, as the function to begin Arduino/ESP8266 communications didn't have any arguments when the function expected parameters. So it appeared to be trying to set their baud rate to 0!!

I edited the library to have a hard value of 57600 and now it seems to work far more reliably.