Jesus gave Peter and the Apostles this same ministry to be carried on in His church. As Catholics we believe there is no such thing as a private sin.
Every sin has a communal effect.
Through the words of the priest we hear for sure and in a public way the forgiveness we seek. We are brought out of our private subjective world into the public objective world where our sin has had an effect. We are given the peace of God's mercy and the assurance of its public announcement.