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All SPI devices on the SPI bus will share those three pins (SCK, MOSI and MISO). The devices must have the slave select (SS) on different pins.
Most RC522 libraries allow the user to configure the SS pin to use. D10 is already used by the w5100 on the ethernet shield so, as a suggestion, you should use D8 for the RC522.
It appears the SPI modes are compatible (MSBFIRST and MODE0), so you should be good to go.
It may be a combination of the length and size of the data line wires. I know it sounds a bit counter-intuitive, but you should use the smallest gauge wire for the data lines. It is sometimes not as much the resistance, but the capacitance of the data lines. If you have an o-scope, monitor the signal on the RC522 end of the lines. If you see any rounding or ringing on the lines, you may need to shorten those wires.
Yo tengo un problema similiar, necesito ayuda, tengo conectador la RFID MCR522 y la Shield Ethernet en arduino UNO y otro en arduino MEGA, necesito dejar los pines digitales desde D10 hasta D13 debido a que no me lee las tarjetas RFID, creo que es porque esos pines son reservados para la Ethernet, que puedo hacer ahí.
Quedo pendiente
Muchas gracias