Is it possible and it is allowed for priests, who after all by virtue of their office forgive the sins of others in the name of God, to pronounce forgiveness for their own sins?

Answer: Only God forgives sins. Since he is the Son of God, Jesus says of himself, The Son of man has authority on earth to forgive sins (Mk 2:10) and exercises this divine power: Your sins are forgiven. (Mk 2:5; Lk 7:48). Further, by virtue of his divine authority he gives this power to men to exercise in his name. [CCC 1441]

Christ has willed that in her prayer and life and action his whole Church should be the sign and instrument of the forgiveness and reconciliation that he acquired for us at the price of his blood. But he entrusted the exercise of the power of absolution to the apostolic ministry which he charged with the ministry of reconciliation (2 Cor 5:18). The apostle is sent out on behalf of Christ with God making his appeal through him and pleading: Be reconciled to God. (2 Cor 5:20). [CCC 1442]

302. What are the essential elements of the sacrament of Reconciliation? The essential elements are two: the acts of the penitent who comes to repentance through the action of the Holy Spirit, and the absolution of the priest who in the name of Christ grants forgiveness and determines the ways of making satisfaction.

307. Who is the minister of this sacrament? Christ has entrusted the ministry of Reconciliation to his apostles, to the bishops who are their successors and to the priests who are the collaborators of the bishops, all of whom become thereby instruments of the mercy and justice of God. They exercise their power of forgiving sins in the name of the Father and of the Son and of the Holy Spirit.

309. Is a confessor bound to secrecy? Given the delicacy and greatness of this ministry and the respect due to people every confessor, without any exception and under very severe penalties, is bound to maintain the sacramental seal which means absolute secrecy about the sins revealed to him in confession. [Numbers 302, 307 and 309 are an extract from the “Compendium of the Catechism of the Catholic Church”. www.vatican.va/roman_curia/pontifical_councils/justpeace/documents/rc_pc_justpeace_doc_20060526_compendio-dott-soc_en.html ])

From this is it evident that all the faithful, including also the pope, bishops and priests, are called to confess their mortal, and preferably also their venial sins, to a priest, in order to receive forgiveness.