Mrs. Peters would be more concerned at first with the legalities of the case against Mrs. Wright because she is the sheriff's wife rather than, like Mrs. Hale, a farmer's wife. Mrs. Peters would know it was important to her husband's career and status to find the killer in a murder case.
Mrs. Peters, as we find out, is also not from the area, as Mrs. Hale is, but a recent transplant. She did not know Minnie Wright as a bright young woman and has not been able to see what happened to her after her marriage.
Mrs. Peters's empathy is engaged only as she gets to know Minnie better by learning about her through Mrs. Hale. As she hears about Minnie's isolation, she is able to relate it to the isolation she experienced on a farm out west. More universally, as a domestic woman, she can understand what it is like to lead a life of cooking, canning, cleaning, and sewing: this is a world that is completely familiar to her, making it possible to walk in Minnie's shoes. Finally, Mrs. Peters can empathize strongly with a male killing a pet and the rage that can cause. As a child, she had to watch helplessly when a boy hacked her kitten to death.
From the start, Mrs. Hale feels differently because she has known Minnie for so long, knows Minnie's coldhearted husband, and understands Minnie's situation. She also has no vested interest in law enforcement, as her husband is a farmer. She contributes to the play by being able to fill in Minnie's backstory. Together, the two women's different experiences bring a complete picture of what Minnie suffered. Mrs. Peters's initial hesitation shows how our opinions of the law can change as we gain more knowledge of a situation.