One consequence of sin in The Scarlet Letter is that the sinner acquires the ability to sense or recognize the sins of others.Hester realizes that...

I don't think that Hawthorne intended for Hester's being able to "see" inside sinners' hearts to be a general "consequence" of sin, applicable to all characters in then novel. Hester is the only character who "sees" and is then sympathetic toward the sinner. In contrast, Chillingworth recognizes the sin in Dimmesdale's heart because he hounds him and notices how Pearl acts around the minister.  He has keen observation skills, but Hawthorne does not portray Chillingworth as being able to look inside all sinners' hearts. Similarly, if Dimmesdale--as a consequence of his sin--had insight into the hidden sins of others' hearts, he would not need Hester to warn him about Chillingworth's intentions toward him; he would have recognized that for himself. Finally, Mistress Hibbens possesses similar "powers" as Hester.  She points and yells at other "sinners" in the novel, but Hawthorne never makes it clear if she is truly a sinner like Hester or if her eccentricities coupled with her psychic-like abilities simply cause others to view her a witch and unrepentant sinner.

More than likely, Hawthorne bestows Hester's unusual skill upon her to demonstrate that a repentant heart has the ability to empathize with and perhaps even counselor others who have sinned.