It is my understanding and belief the soul is composed of good and evil but it is one entity. The determination of where the soul goes after the body dies is whether the person has made reconciliation with God for their sins. If their peace is made with God the soul goes to Heaven. If no peace is made with God the soul goes to hell.
"Papal teaching" would specifically refer to teachings of the Pope. Not being Catholic, I cannot reliably speak to what the Pope does or does not teach. However, I have studied the Bible and I can speak to what the Bible teaches.
Humans were created in their entirety to be in fellowship with God. Sin broke that fellowship, but Jesus paid the price for that break and offers reconciliation with God. If we have accepted that salvation, if a person is reconciled with God, their soul--their entire soul--will go to heaven and live eternally with God. There is no "good side" and "bad side" to the human soul. We are one being, created to be perfect but allowed to make imperfect choices. A redeemed soul will stand as perfect before God because Jesus' saving grace will cover those blemishes. Hell is not meant for human souls, but those who are not redeemed, those who choose to live in opposition to God will go there when they die--their whole soul will go there, even if they were a "nice person." The Bible does not teach or suggest that the human soul is split with a good part and a bad part. In fact the Bible says we "cannot serve two masters..." Our souls are one, we are one entity. Sometimes we make good choices, sometimes we make bad choices, in the end the destination of our soul is determined by whether we have accepted Jesus' redemption and are reconciled with God or not.