Hester finds peace through the acceptance of her own humanity with all its limitations. Hester accepts the punishment of the community because she knows that she has violated one of their regulations; violation of the legitimate regulations of any community can result in legitimate punishment. Her often quoted line, by me at least, "What we did had a consecration of its own" explains how she has made peace. Unlike Dimmesdale, who is supported by an "iron framework" of belief that, while supporting him confined himself in his own prison, Hester believes that what they did, while wrong in the eyes of the community, was "consecrated," a word with very specific religious connotations.
Hester finds peace within herself, with her knowledge or belief that there was something "sacred" (consecrated) in what they had done. This forgiveness is beyond Dimmesdale, and because of this he will never experience the peace that Hester found.