The term "sin" is inherent to the premise that one believes certain acts as moral or immoral. In order to base this premise one has to truly believe that there is validity to the concept of morality or immorality. Without that validation, the concept of sin becomes null.
Those who are adepts to the validity of morality over immorality are who truly believe in a concept such as sin which, in their eyes, is the commission of an act that breaks with an established paradigm based on moral rules.
Analyzing from a contrasting perspective, it is safe to say that those who do not believe in any paradigm based on a moral code also do not believe in the concept of sin. Since everybody is equally entitled to the right of believing in what they want, then it would be a silly and unnecessary, not to mention impertinent to suppose that someone must be held accountable in public for the commission of something in which many would not see as fault.
In The Scarlett Letter the society of Hester Prynne is homogenous and cohesive in terms of creed. For this reason, there is no doubt that they all would agree in submitting Hester to the public humiliation of exposing her sin to the censure of the settlers. However, this is unique to their specific type of population, not to mention to their particular tendencies to use tactics such as the scaffold as a way to bring on shame. This is their way to play God and punish the sinner. However, is it really punishing the sin? Moreover, is there such a thing as sin? It is obvious that, to Hester, her act may have felt and looked as a sin once, since she was forced to hide herself to commit it. It is her pregnancy what makes her sin take shape, form, and consequence.
When it comes to Hester’s society, the exposure of sin to public censure was a common practice based on their particular system of belief. However, the modern reader can see how an abstract and circumstantial idea such as that of “sin” is completely inherent to the believer. Therefore, it makes no difference whether it is put to public censure or not. In the end, it all depends on whether you believe in having committed it, or not.
Before addressing The Scarlet Letter directly, let us remember two Biblical admonishments: “Render to Caesar the things that are Caesar’s, and to God the things that are God’s” and “Let him who is without sin cast the first stone.” Hawthorne was deliberately novelizing a special case, one where religion and social behavior were close interwoven (and the book is as much about pride and about vengeance as it is about sin). The humiliation of Hester Prynne was only part of the punishment; her public “badge” was also a signal to other citizens to avoid her company. And that is where sin and public censure intersect: if the individual’s sin has a direct effect on the public’s welfare, it must be recognized by that society. Therefore, sins that abuse public office should be censured, but sins that simply are part of the human imperfections need not be. The confusion enters into the equation when someone claims that an individual’s “sinning” behavior is prima facie evidence that he or she is unfit for public office, because no-one would be “qualified” if we waited for a sinless person. The public gleefully judges the behavior of others when some connection can be made between private and public behavior, on the supposed grounds that we all are the arbiters of our public officials’ morality. In The Scarlet Letter, the mise-en-scene calls for all behavior to be judged publically, but only Hester Prynne’s behavior is publicly censured, not Arthur Dimmesdale’s. Hopefully we have moved a little away from that condition.
Although this question is also based on personal opinion, let's try to be as objective as we can.
In Nathaniel Hawthorne's The Scarlet Letter the exposure of sin to the public eyes of the villagers is a strategy used by puritans and Calvinists to instill fear, humiliation, and guilt in the population, hence, motivating them to not commit the same sin, themselves.
Meagre, indeed, and cold was the sympathy that a transgressor might look for, from such bystanders, at the scaffold. On the other hand, a penalty, which, in our days, would infer a degree of mocking infamy and ridicule, might then be invested with almost as stern a dignity as the punishment of death itself.
However, the novel shows us that this practice is quite hypocritical. Hester's village is rife with sin. The "goodwives" talk behind Hester's back, the villagers are judgemental, Mistress Hibbins is a witch, and Dimmesdale is a liar. This latter fact is the most important: Dimmesdale is the co-author of Hester's so-called crime. He is her lover, and he is the father of her daughter. And, yet, he stands at the scaffold (albeit, with less severity) demanding to extort Hester's "truth" because he is expected to do so by a community who, erroneously, admires him.
The Reverend Master Dimmesdale, her godly pastor, takes it very grievously to heart that such a scandal should have come upon his congregation.
This being said, we should ask ourselves: who are we to judge? Who is really free of sin to the point of giving themselves the task of judging the sins of others? Nobody, at all, has the right to lift one finger to point another person, nor say one word against someone else. For this reason, Hawthorne uses his novel to tell us the degree of hypocrisy that can inhabit within one same, so-called "good" community. Therefore, there should never be any reason to publicly point out the doings of other people, unless you are ready to suffer the same process, yourself.
I cannot help but feel that Hawthorne would suggest that the issue of sin and redemption are subjective voyages within the individual. They are complex and intricate, impossible to externally quantify and completely dependent on the individual and their desire to find internal peace between themselves and their vision of the divine. Dimmesdale and Hester would demonstrate this, as their voyages towards spiritual redemption are so private and internal that the public could not begin to understand or comprehend it. In this, I think that Hawthorne would suggest that sin should not always be exposed to public censure. The idea of sin and sinful behavior being so public and so subject to the realm of the external seems to be the exact behavior that he is criticizing. Hawthorne might be suggesting that there is so much ambiguity in the notion of spiritual worship that individuals cannot expose this to public scrutiny, a domain where so many forces conspire to utilize the idea of spiritual redemption as a means to an end as opposed to an end in its own right. Through this, I think that Hawthorne would reject the idea that sin should be exposed to public censure.