I read this story at least 20 years ago and almost certainly it is quite a bit older than that. I would hazard a guess that it is 1980's or earlier, but don't actually have a clue. I don't remember the cover art. I think it was a short novel or novelette.
I could have sworn it was by Margaret Atwood, but nothing in her oeuvre is ringing any bells. It could also have been by one of the classics of SF, such as Heinlein (again no bells), but also possibly not. I likely got the book from my local library.
Set in a prosperous future after an apocalyptic event, or may start pre-apocalypse with a description of society at the time. The main narrator tells their tale from the first person perspective I think, describing a number of events, such as going for a ride on their favourite riding beast and participating in races on the beast. They also describe some of the difficulties of breeding the beasts (like all breeding a racehorse to us) - they are fiercely independent in this and will mate indiscriminately, but only with their established partner, not one chosen by their owner (if that makes sense).
The animals are described as making sounds that almost make sense to the owners and seem like they are communicating with each-other, but are described as "dumb animals" by the narrator.
Long story short; it turns out that the "beasts" are humans, who chose to atone for their damage to the environment (hence the Atwood in my mind), and to do this they deliberately turned themselves into servants for other animals, helping them to evolve along better lines than humans did over many years, possibly using genetic modification to speed the process. The humans have also taught the animals English as their language, including written as well as speech. The sounds that the humans are making are indeed language, possibly some evolution of pig-latin, used deliberately to ensure that their animal (now) masters can't understand them and allow the humans to continue their assistance of the evolutionary path.
I think the narrator of the story is actually a descendant of a squirrel or some similar animal, but am very uncertain about this.