Objection 1. It would seem that a penance should not be published or solemnized. Because it is not lawful for a priest, even through fear, to divulge anyone's sin, however notorious it may be. Now a sin is published by a solemn penance. Therefore a penance should not be solemnized.
Objection 2. Further, the judgment should follow the nature of the tribunal. Now penance is a judgment pronounced in a secret tribunal. Therefore it should not be published or solemnized.
Objection 3. Further, "Every deficiency is made good by penance" as Ambrose [Cf. Hypognost. iii, among the spurious works ascribed to St. Augustine] states. Now solemnization has a contrary effect, since it involves the penitent in many deficiencies: for a layman cannot be promoted to the ranks of the clergy nor can a cleric be promoted to higher orders, after doing solemn penance. Therefore Penance should not be solemnized.
On the contrary, Penance is a sacrament. Now some kind of solemnity is observed in every sacrament. Therefore there should be some solemnity in Penance.
Further, the medicine should suit the disease. Now a sin is sometimes public, and by its example draws many to sin. Therefore the penance which is its medicine should also be public and solemn so as to give edification to many.
I answer that, Some penances should be public and solemn for four reasons. First, so that a public sin may have a public remedy; secondly, because he who has committed a very grave crime deserves the greatest confusion even in this life; thirdly, in order that it may deter others; fourthly, that he may be an example of repentance, lest those should despair, who have committed grievous sins.
Reply to Objection 1. The priest does not divulge the confession by imposing such a penance, though people may suspect the penitent of having committed some great sin. For a man is not certainly taken to be guilty, because he is punished, since sometimes one does penance for another: thus we read in the Lives of the Fathers of a certain man who, in order to incite his companion to do penance, did penance together with him. And if the sin be public, the penitent, by fulfilling his penance, shows that he has been to confession.
Reply to Objection 2. A solemn penance, as to its imposition, does not go beyond the limits of a secret tribunal, since, just as the confession is made secretly, so the penance is imposed secretly. It is the execution of the penance, that goes beyond the limits of the secret tribunal: and there is nothing objectionable in this.
Reply to Objection 3. Although Penance cancels all deficiencies, by restoring man to his former state of grace, yet it does not always restore him to his former dignity. Hence women after doing penance for fornication are not given the veil, because they do not recover the honor of virginity. In like manner, after doing public penance, a sinner does not recover his former dignity so as to be eligible for the clerical state and a bishop who would ordain such a one ought to be deprived of the power of ordaining, unless perhaps the needs of the Church or custom require it. In that case such a one would be admitted to minor ordersby way of exception, but not to the sacred orders. First, on account of the dignity of the latter; secondly, for fear of relapse; thirdly, in order to avoid the scandal which the people might take through recollection of his former sins; fourthly, because he would not have the face to correct others, by reason of the publicity of his own sin.
Source: Summa Theologica (New Advent)