This is the essence of confession (a renewal of baptism). It was originally a practice that started with ordinary people seeking out desert monastics who spent their live in askesis. Only later was it usurped by priests whose actual job was just the rite of absolution.
The belief of the Greek Church is naturally also that of the Russian in this regard. Russian Orthodox theologians all hold that the Church possesses the power to forgive sins, where there is true repentance and sincere confession. The form in use at present is as follows: "My child, N. N., may our Lord and God Christ Jesus by the mercy of His love absolve thee from thy sins; and I, His unworthy priest, in virtue of the authority committed to me, absolve thee and declare thee absolved of thy sins in the name of the Father, and of the Son, and of the Holy Spirit, Amen."
