What is Hester's demeanor as she emerges from prison in The Scarlet Letter?

It depends on which occasion you are referring to. In the beginning, she merges to public scrutiny  with a quiet dignity and defiance. She retreats into memories of her past to deflect the humiliation she must endure. The second time, she is much less composed. She is worn down, and appears to suffer the consequences much more openly .

What is Hester's demeanor as she emerges from prison in The Scarlet Letter?

Hester assumes her humanity rather than fight against it; in many ways, her “sin” derived from  her acknowledgment of her human need for love, following her husband’s unexplained failure to arrive in Boston and his probable death. She is criticized by the  women of the town for her elaborate embroidery of her scarlet letter, the symbol of her shame: its ornate design  seems to declare that she is proud, rather than ashamed, of her sin. In reality, however, Hester simply accepts the “sin” and its symbol as part of herself, just as she accepts her child. Although she can hardly believe her present “realities,” she takes them as they are rather than resisting them or trying to atone for them.

What is Hester's demeanor as she emerges from prison in The Scarlet Letter?

Hester has been in prison for three months. At first, she pulls the baby closer to her, as if ashamed, but then she walks with grace and dignity to the scaffold to endure three hours of public humiliation. Her walk shows how strong her character is, as she raises her head in pride and displays a "burning blush" and a "haughty smile". Her actions are "marked with natural dignity and force of character, and stepped into the open air as if by her own free will."  Hester is tall, "with a figure of perfect elegance on a large scale." People are amazed at her beauty, and she "was ladylike, too, after the manner of the feminine gentility of those days; characterised by a certain state and dignity, ..." The people who knew her before she went to prison were amazed at her appearance because she seemed to make "a halo of the misfortune and ignominy in which she was enveloped." Hester looks more like a woman of nobility than a prisoner and seems not to have physically suffered as a result of her imprisonment. She "seemed to express the attitude of her spirit, the desperate recklessness of her mood, by its wild and picturesque peculiarity." Hester wants to show the onlookers that she loves her baby and plans to raise her as best she can.

What most noticed about Hester, however, was "that SCARLET LETTER" , and the people were spellbound by it.

How does Hester behave when she returns to prison in The Scarlet Letter?

After her ignoble exposure upon the scaffold, Hester Prynne is returned to the weather-beaten and rusty prison, that bastlon of Puritan punitive measure.  Immediately, she becomes as still as a corpse, although her babe continues to whimper. There also is in the prison the stranger whom she has spotted in the crowd, the man who has been with the Indians, Roger Chillingworth. This man is her husband. And, when he offers to treat the restless baby, Hester mistrusts him as she is in "no reasonable state of mind." 

Chillingworth then examines Hester herself; as he is calm and intense, Hester reacts to his gaze by shuddering and shrinking back from him, worried what his intentions are. When he hands her a cup from which she is to drink, Hester hesitates. 

"...if death be in this cup, I bid thee think again, ere thou beholdest me quaff it.  See! it is even now at my lips."

Chillingworth assures her that he will derive more satisfaction in knowing that she wears the scarlet letter. He also informs her that he might have "beheld the bale-fire of that scarlet letter blazing at the end of our path," meaning that because he is older than Hester and a "man already in decay," she would look for someone who could share her passion. To this accusation, Hester replies that she was honest with him, having no love for him. Further, Chillingworth asks Hester with whom she has sinned, but she refuses to reveal the man's name. Undeterred by her refusal, Chillingworth declares, "Not the less he shall be mine!" A bewildered Hester wonders why he has helped her, but threatens the man with whom she has sinned; in addition, he threatens Hester if she should reveal his own identity. These words frighten Hester Prynne.