We’re getting right off the subject.
What's the role of the 'right'? Does it stress the meaning of the sentence?
We’re getting [right off the subject]
"Right" is an adverb here.
It's a modifier in the structure of the bracketed preposition phrase.
Here, it metaphorically denotes exactitude of direction.
It stresses the extent to which -- i.e. how far -- we are getting away from the initial subject. But specifically, it does that in such a way as to imply that we shouldn't be doing it!
So, yes, in that sense it emphasizes the meaning.
COMPARE:
right means straight: right to the point or subject and that means directly or exactly to the point or subject.
Generally, when going off course as regards a topic, adjectives meaning precisely are not used.