Hester refuses to announce the name of her co-adulterer for many reasons. She is unable to deny her lack of faithfulness to her marriage vows because of the evidence of her pregnancy but her lover can remain anonymous. Through her refusal, she may be trying to force her co-adulterer to confess his own sin rather than have her do it for him. Also, if her co-adulterer were to come forward, there is the potential for both Hester and he to be executed under Puritan law for the crime of adultery leaving Pearl to be the ward of a community which is predisposed to be hostile to her. Also, if Hester announces the name of her co-adulterer, there is no potential for the two to somehow eventually legitimize their sin through marriage. Hester later suggests to Dimmesdale, her co-adulterer, that they leave their community and the "shame" of the scarlet letter behind and live together as man and wife with Pearl as their legitmate daughter. Essentially, Hester "denies her fellow-sinner" in order to protect both Pearl and Dimmesdale. She is willing to bear the penalty for the sin alone so that she can raise Pearl herself and so that she can retain the hope that something good might eventually come of the situation.
The irony of this scene is crucial to its meaning, because, of course, Dimmesdale is asking Hester to speak his own name as the “fellow-sinner,” which she knows, we soon learn, but the community doesn’t know at all. Furthermore, while he pleads with her to denounce him because he is to weak to admit his guilt, Hester has the opportunity to force him to experience the guilt he will not admit. “Deny” has several meanings in the Oxford English Dictionary, each of which complicate the passage. It can mean “to refuse to admit the existence of,” “to refuse to admit or acknowledge,” and also, interestingly, “to disown, disavow, repudiate, renounce. Each meaning, in the context of this passage, becomes more pejorative, suggesting the hostility in Hester’s refusal: she refuses to say he exists, and she refuses to save his soul as well.
In chapter three, "The Recognition", Reverend Dimmesdales pleads with Hester to reveal the name of her "fellow-sinner", or co-adulterer. When he pleads with Hester not to deny this person, Dimmesdale is arguing that it would be better for this man if Hester would reveal his name. Dimmesdale tries to persuade Hester by telling her that it is worse for this man to live with the guilt for the rest of his life than it would be for him if Hester just revealed his name. Dimmesdale claims that the man might not have the strength to come forward himself, but she would be doing the co-adulterer a favor by confessing his name for him. By keeping quiet, Dimmesdale claims that Hester is 'denying' this man from the freedom that comes from confessing his sin.