A coyote may follow you and your dog — the dog is the issue — out of curiosity or to monitor it, the same way you yourself might follow a “suspect” prowling through your neighborhood, to find out where they were going and what they were doing.
If you find that you are being followed by a coyote, walk away from the coyote — and don’t run, running invites chasing. Keep aware of the coyote and shoo it off effectively if it gets too close, and move on. And keep your dog leashed. Pick up a small dog.
The leashing is to keep your dog from being distracted by the coyote and going after it. You want to avoid engagement between the two.
I’ve seen this same following-behavior used for a purpose totally different from either curiosity or monitoring. It was used effectively by a coyote to avoid detection, as a human and his dog passed by. The dog had a history of chasing the coyote, and the man had a history of pursuing the coyote aggressively with his camera. So this coyote had a particular interest in avoiding this duo. The dog and person passed while the coyote stood absolutely still and remained hidden and undetected in a dark wooded area. Then, to my great surprise, the coyote came out of hiding and followed them at a close 30 feet. The coyote did so carefully, on high alert and prepared to bolt if necessary. This went on for about 200 feet before the coyote veered off to where the brush picked up again and it could continue undetected through the bushes. Neither the man nor his dog ever looked back!
In this case, what seems to be going on is that, by following in the duo’s “wake”, the coyote was continuing to avoid detection. Animals and people tend to look around themselves, but much less frequently directly in back of themselves. We all tend to concentrate on sounds, smells and sights which are in front of us or to the sides. Coyotes know this, and “follow” as a method to avoid being seen.