You can access Mediator Dei on the Vatican web site.
It might be that the following error was more prevalent in 1947:
83. For there are today, Venerable Brethren, those who, approximating to errors long since condemned[Cf. Council of Trent, Sess. 23. c. 4] teach that in the New Testament by the word “priesthood” is meant only that priesthood which applies to all who have been baptized; and hold that the command by which Christ gave power to His apostles at the Last Supper to do what He Himself had done, applies directly to the entire Christian Church, and that thence, and thence only, arises the hierarchical priesthood. Hence they assert that the people are possessed of a true priestly power, while the priest only acts in virtue of an office committed to him by the community. Wherefore, they look on the eucharistic sacrifice as a “concelebration,” in the literal meaning of that term, and consider it more fitting that priests should “concelebrate” with the people present than that they should offer the sacrifice privately when the people are absent.
On the other hand, it strikes me as a better expression to refer to baptized believers as celebrating Mass, though in a different way from those in Holy Orders.
Regarding the “private Mass,” there’s a long tradition behind it. It doesn’t quite go back to antiquity, and some council bishops at Trent saw it as a stronger bulwark against error and mishap.
84. It is superfluous to explain how captious errors of this sort completely contradict the truths which we have just stated above, when treating of the place of the priest in the Mystical Body of Jesus Christ. But we deem it necessary to recall that the priest acts for the people only because he represents Jesus Christ, who is Head of all His members and offers Himself in their stead. Hence, he goes to the altar as the minister of Christ, inferior to Christ but superior to the people.[Cf. Saint Robert Bellarmine, De Missa, 2, c.4] The people, on the other hand, since they in no sense represent the divine Redeemer and are not mediator between themselves and God, can in no way possess the sacerdotal power.
The superiority is not a moral one, but one defined only by the ritual offered for the people. If there is any “superiority” it is not due to the action of any human being, but to the grace of Christ.
85. All this has the certitude of faith. However, it must also be said that the faithful do offer the divine Victim, though in a different sense.