In what circumstances do departments permit graduate students to complete their program off campus?
This is really something that will differ depending on where you are, what field, and even who your advisor is.

For example, at my previous institution, the rule was that you couldn't work on your dissertation remotely until after your proposal date (meaning you are fairly far into your program - proposals at that institution were 2-3 years in for many) and only with your committee's approval. Some chairs could, and did, refuse.

Others were extremely laid back, and probably wouldn't have objected to skirting that rule.