MFRC522 RFID writing personal data
There is no read back verification. The relevant parts of Mifare_Write():

// Step 2: Transfer the data
result = PCD_MIFARE_Transceive(buffer, bufferSize); // Adds CRC_A and checks that the response is MF_ACK.
if (result != STATUS_OK) {
    return result;
}

return STATUS_OK;


The data is merely transferred to the reader's FIFO and then an ACK is expected within a timeout period; the ACK is the most confirmation you'll get from the reader that your data was successfully transferred. If you want anything extra, you'll have to read back the blocks with Mifare_Read() and conduct your own verification. 

Of course, all this assumes you're using this library by Miguel Balboa, though I strongly doubt you'll get different results with other libraries; there's just no point in including this function in the library when the user can easily do it if they choose to.