No. The priest cannot require the penitent to reveal his/her sins to a third party. The only possible exemption is if the penitent is confessing to having falsely accused a priest of soliciting him/her in the confessional, in this set of circumstances the penitent is required to publicly withdraw the accusation before absolution can be given (canon 982). Outside of that extremely rare situation, the confessor cannot require anyone to reveal their sins to another.