The answer is yes, he does. Te priest will tell the penitent that the priest has been given the authority to inform the penitent that his sins have been absolved by God.
Very well; therefore, the Orthodox priest does make a decision to absolve or not that is efficacious; whether he verbalizes it with "absolvo te" or with some other language, or not at all, is a secondary matter.
It is not like "filioque" where the procession of the Holy Ghost is a real theological contention. It is more like the Orthodox epiclesis as opposed to the Catholic institution where the form is different but the theology is the same, and the supernatural effect is the same.