you have 3 devices for the RX/TX which can connect only two devices in booth directions. your shield connects to Atmega RX/TX which is connected to USB RX/TX too. but the esp8266 is to USB connected TX to TX and RX to RX so it can't communicate with Serial Monitor in this configuration. with a module one simply swaps RX/TX connection of esp8266 to Uno to communicate from esp8266 to Serial Monitor. has the shield some switches?
USB TX -> Atmega RX <- esp8266 TX
USB RX <- Atmega TX -> esp8266 RX