how do you power the RC522? I connect
ESP8266 GND to RC522 GND
ESP8266 3.3V to RC522 3.3V (the power pin on the RC522 is labeled 3.3V)
the LED on the RC522 is then bright
could your RC522 require 5Volts? if so you then require a level converter to connect it to the ESP8266
see @UKHeliBob comment in post 2