Hester found peace because she was dealing with something that required acceptance of external punishment rather than dealilng with the intense blame and guilt of Dimmesdale. In fact, Hester becomes a vital part of the community, often helping others who are in need --- this despite the fact that her good work is not appreciated.
Thus it was with the men of rank, on whom their eminent position imposed the guardianship of the public morals. Individuals in private life, meanwhile, had quite forgiven Hester Prynne for her frailty; nay, more, they had begun to look upon the scarlet letter as the token, not of that one sin for which she had borne so long and dreary a penance, but of her many good deeds since. "Do you see that woman with the embroidered badge?" they would say to strangers. "It is our Hester--the town's own Hester--who is so kind to the poor, so helpful to the sick, so comfortable to the afflicted!" Then, it is true, the propensity of human nature to tell the very worst of itself, when embodied in the person of another, would constrain them to whisper the black scandal of bygone years. It was none the less a fact, however, that in the eyes of the very men who spoke thus, the scarlet letter had the effect of the cross on a nun's bosom. It imparted to the wearer a kind of sacredness, which enabled her to walk securely amid all peril. Had she fallen among thieves, it would have kept her safe. It was reported, and believed by many, that an Indian had drawn his arrow against the badge, and that the missile struck it, and fell harmless to the ground.
Although this judgment appears at the end of the book, I am certain that it's something that Hester felt throughout. Unlike Hooper in The Birthmark, Hester was able to take off the letter, at least in the forest with only her family present. She put it on when she returned to the world and, of course, to bring reality back to Pearl who didn't recognize her without it ... for practical and symbolic reasons. Even when she returns from the Old World, she puts the letter back on, even though no on would have minded if she had not.
Of course she lived a life without the man she loved, and that must have taken some of the happiness from her life, but her internal compass was always directed by her own words: "What we did had a consecration of it's own! We felt it!"
I think Hawthorne does want the reader to know that Hester has found a sense of peace through her punishment. The letter has forced her to be independent and to think for herself. With her sin being out in the open (i.e, having to wear the letter, serve prison time, and stand on the scaffold), Hester has not been tortured by her guilt as Dimmesdale has for years. The letter allows her to act freely in a sense because others have low expectations of her. However, by not fulfilling their low expectations, Hester not only "gets the better of them," but she also feels a sense of self accomplishment. She establishes herself with a needed skill and truly ministers to others in their time of need.