The Jewish Model
The High Priest, Aaron & Successors
The common Priesthood, Levites
The Priesthood of Believers, Israelites and Jews
The Christian Model
The High Priest, Jesus Christ
The Common Priesthood, Bishops and Priests
The Priesthood of Believers, Christians
In both models, there is no priesthood without sacrifice.
In the Jewish priesthood structure, for the most important sacrifices, the High Priest would make the sacrifice, often with the assistance of common priests, who would lend their hands to his efforts, which were made with the prayers and on behalf of the priesthood of the faithful for the entire nation.
For less important sacrifices, like an individual purification right, the Levite priest would make the sacrifice with his hands, in the authority of the high priest, with the prayers of the family who the sacrifice was made for.
In the Christian priesthood, the intended structure is similar. Jesus is the High Priest, with a common priesthood of bishops and priests, who cooperate with Him and us, the priesthood of believers, to make the sacrifice of the Mass his continuing power and presence working in the world. Jesus makes the real sacrifice on his altar in heaven on our behalf, where we will participate when we get to heaven. Here on Earth, the common priests lend their hands and voices to his cause, so he can make his sacrifice present to us for our benefit, as we join our prayers to his efforts.
The Protestant rejection of this teaching rejects that there is a sacrifice in the new covenant. Unfortunately, that ignores the fact that there is no priesthood without a sacrifice. By the very fact that there is a priesthood, there must be a sacrifice. If the Protestant model were true, then there would barely be a High Priest, Jesus, who sacrificed himself once upon a time, but there wouldn't be an ongoing sacrifice that makes the power of his death and resurrection active in our lives today. Also, with no ongoing sacrifice, there would be no priesthood of believers, which the bible clearly teaches there is.
In Hebrews 7, we see that Jesus is a priest like Melchizedek. The priesthood of Melchizedek is significant because he made an offering of bread an wine (Gen 14:18). This is not negated by Hebrews 7: 27 (Unlike the other high priests, he does not need to offer sacrifices day after day, first for his own sins, and then for the sins of the people. He sacrificed for their sins once for all when he offered himself.) This is true, he does not make are the sacrifices of the old law like the Jewish priests. He does offer us a sacrifice of himself in the bread and wine, which is not "sacrifices", but a singular sacrifice, by which he maintains his power and presence in the world, so he is still Immanuel, God is with us. Without this sacrifice for us to participate in, there is no priesthood for us to be a part of.