Hester fears for Dimmsdale and tells him not to say anything about his part in adultery. He responds by wanting to come forward but I believe internally he is grateful that he does not have to tell the truth. Hester believes that he is supposed to be the good minister that he is for the people and does not want him to suffer just because she will have to suffer.
Dimmsdale allows Hester to stand on the scaffold by herself and face her accusers but he says nothing. Yet, he also must be agonizing that at some point she may tell on him. However, Hester's character is such that she will not, but rather waits patiently.