What are the 2 contrasting views of Chillingworth and his relationship with Dimmesdale that are held by the residents in The Scarlet...

Because the Puritans believe that everything is part of "God's plan," the members of Hester's community, especially the religious elders, are convinced that the physician's arrival and his apparent solicitude toward the minister is an act of Providence. However, Hester, who is not a Puritan, views Chillingworth's arrival as anything but providential and opportune, aware of his secret plan to destroy the minister.

In Chapter IX when Chillingworth's knowledge of medicine becomes known, "a rumor gained ground" within the Puritan community that "Heaven had wrought an absolute miracle" by transporting a Doctor of Physic from a university in Germany to their area. When this doctor expresses interest in the minister, the elders, deacons, and "motherly dames" feel he has been sent by God to attend their beloved minister. For, the Reverend Dimmesdale is clearly ailing. Therefore, the community becomes "importunate that he should make trial of the physician's frankly offered skill." In fact, the elder minister of Boston, and the deacons of his own church confer with him about his sin of rejecting help until Dimmesdale acquiesces and allows the physician into his home.

Certainly, Hester's view is more accurate than that of the Puritan community since hers is founded in reality. In Chapter IV, for instance, Roger Chillingworth visits Hester in the prison and the "book-worm of great libraries,--a man already in decay--" questions Hester about her lover, but she refuses to reveal anything. So, he tells his unfortunate young wife who refuses to answer that there is a "sympathy" between him and the other man that will cause the man's nature to be revealed, anyway:

"I shall see him tremble. I shall feel myself shudder, suddenly and unawares. Sooner or later, he must needs be mine."

After having felt this "sympathy" between him and the Reverend Dimmesdale, the man now perceived by the community as a worthy physician initiates his plan for revenge. While he administers his counsel and applies his knowledge of herbs and cures learned during his captivity with the Indians, Chillingworth assiduously watches the minister, questioning him, and listening to what is not being said.   

One day as he speaks with the minister, the physician tells Dimmesdale, "...the disease is what I seem to know, yet know it not." He says that he seeks what psychological trouble lies behind the physical signs of illness so that he can better treat the minister. But, the fearful minister refuses, "Not to thee!" asserting that only God should know what lies in one's soul. 

"But who art thou, that meddlest in this matter?--that dares thrust himself between the sufferer and his God?"

Nevertheless, Chillingworth does thrust himself between the minister and God, becoming a "fiend," a name he even gives himself when he talks with Hester later. He becomes this fiend shortly after his prying conversation with the minister. For, while Dimmesdale sleeps, the physician pulls open his clothing on his chest and discovers the sign of the minister's sin on the skin.

But what distinguished the physician's ecstasy from Satan's was the trait of wonder in it! (Ch. X)

Hawthorne's development of character, as well as Hester's true knowledge of Roger Chillingworth, point to the real nature of the physician. Clearly, he is not the intermediary from Providence that members of the Puritan community believe him to be.