'What's that Skip?' Researchers say kangaroos can communicate with people

Evolution has occurred over the past 3.5 billion years, and, relatively speaking, our common ancestor isn't all that distant. There could also be some convergent evolution. We're very different from elephants, but our brains are also similar in a lot of ways. There's "something it's like to be an elephant" that we can never know, but which might not be so drastically different from our own experiences that we couldn't ever envision it.

That feeling we experience when we socialize with each other might very well be close to the feeling they experience, too. It may generate the same neurotransmitter activity, the same subjective emotions, maybe the same feeling of empathy. Perhaps not, but it's certainly very plausible, especially if you observe them for a long period of time. Who knows, maybe it's even near-identical in some regards.

I also remember Fury the horse and Flipper the dolphin in the same genre.

There's an attempt to refine the normal dog-human interpretative dance into a more formal setup.

Some people are using buttons to get the dog to reply in a context format rather than just gauging enthusiasm for a given task.

https://www.hungerforwords.com

I don't think higher order communication between adults is that simple either. There's a lot of inference and deriving from implication that is prone to error.

However, there's plenty of evidence that various species have cracked human communication, at least partly so, while we haven't that much to show…

Let's say, all those where some individuals manage to adopt human speech, like various birds, some whales, chimps, etc, or manage to interact with human language symbols by means of sign language or similar. This is quite impressive. To my knowledge, we, while far better equipped, haven't intruded a single communication system of an other species in a way that we could meaningful interact or even use their intrinsic concepts. On a less complex scale, but still impressive are those, who manage to interpret some human practices, like pointing. (Mind that this requires some kind of mirroring to construct an assumed line of sight and some assumptions about what the other might have in mind and what might be an object of interest and engaging in some sort of dialog over this.)

dogs

Anecdotally, We got a couple of Bengal cats a few years ago. We got into the habit of saying 'Hello <cat-name>' every time a cat walked into the room.

Now the cats say 'eh-owww' whenever they see us, and also say the same phrase in different ways depending on what they want/need. It took us a while to work out the meanings of the intonations, but now we are pretty good at working out if it's a greeting, request, or just acknowledgement.

I'm not sure who trained who, but it seems to work. :D

It could also be that interest in communication is generally an "uphill game". Which may not be that fortunate in case of contact with an advanced civilization:

We: "Hey, we want to share our science and philosophy and what we found out about the universe!"

Alien scientist (looking at some advanced gauges): "Look, its temperature rises."

;-)