In the Scarlet Letter Chillingworth is living with Dimmesdale because his health recently became bad. But whose idea was it and why?
The people of Boston believe that the divine must have intervened to bring Roger Chillingworth to the town, and to Mr. Dimmesdale, especially in the clergyman's time of apparent medical need. In fact, a rumor begins to circulate
that Heaven had wrought an absolute miracle, by transporting an eminent Doctor of Physic from a German university bodily through the air and setting him down at the door of Mr. Dimmesdale's study!
Although not absolutely everyone believed in this fortuitous coincidence of events as an example of the divine's involvement in their lives,
The elders, the deacons, the motherly dames, and the young and fair maidens of Mr. Dimmesdale's flock, were alike importunate that he should make trial of the physician's frankly offered skill.
At the very least, then, everyone in the town -- whether they believed Chillingworth's appearance to be miraculous or not -- encouraged a close relationship between the reverend and the doctor. Finally convinced by these individuals to consult with Chillingworth, Dimmesdale meets with the doctor, and, over time, the two begin to spend a lot more time together. They visit one another at home, and they got to know and appreciate one another's perspectives. Then,
After a time, at a hint from Roger Chillingworth, the friends of Mr. Dimmesdale effected an arrangement by which the two were lodged in the same house; so that every ebb and flow of the minister's life-tide might pass under the eye of his anxious and attached physician.
Thus, it sounds as though the idea to move in together really originates with Chillingworth but is made possible by Dimmesdale's friends, those who insist on the arrangement for the sake of the minister's health.
Roger Chillingworth, "the good physician," considers Reverend Arthur Dimmesdale's failing health to be to his (Chillingworth's) advantage. However, Dimmesdale's friends (presumably men such as Reverend Wilson and Governor Bellingham) ultimately persuade Dimmesdale to avail himself of the doctor's assistance:
After a time, at a hint from Roger Chillingworth, the friends of Mr. Dimmesdale effected an arrangement by which the two were lodged in the same house; so that every ebb and flow of the minister's life-tide might pass under the eye of his anxious and attached physician (125).
...and of course, Chillingworth would have "hinted" this was the ultimate arrangement he desired! -- for this way, he could constantly keep watch over his patient and could
now [dig] into the poor clergyman's heart, like a miner searching for gold; or, rather, like a sexton delving into a grave, possibly in quest of a jewel that had been buried on the dead man's bosom....(129).
Something -- perhaps Chillingworth's intuition? -- is telling him that he needs to investigate Dimmesdale. What better way to do so than to constantly be with him? Ultimately, you as the reader decide whether or not Dimmesdale's friends make a good decision that actually improves the minister's health.