Or we could be trying to communicate with parrots (and preventing some species of them from going extinct), as suggested in Ted Chiang's short story "The Great Silence".

I suspect that's not going to happen until we have a set of accurate species-specific cognitive models. The hope with SETI is probably more that we can find something smart enough to be able to communicate with us. As someone who lives with a handful of parrots, my experience is that the sort of communication you're talking about is insanely difficult.

It's common for birds to learn a few (human) words in context of an action (names / contact calls, food, waste disposal, etc), and it's common for someone who lives with birds to learn what their different types of calls generally mean (contact calls, distress calls, etc). Interestingly, parrots do seem to exhibit many emotions found in humans (anger, fear, happiness, sadness, love, jealousy, etc -- I suspect these are common among social animals).

If you think of language as a way to externalize our internal state then a prerequisite to understanding and communication is the ability to translate that state into something the recipient can contextualize. The difficulty of communication then increases with the difference between the party's respective inner states.

For human-human interactions it's possible to start from nothing and learn to communicate effectively in a couple of years. For human-parrot interactions, we can literally chatter to each other all day for 50 years and we'll never develop anything more than a cursory grasp of each other's languages.

On a more cynical note, were we able to properly communicate with animals it would force us to confront some uncomfortable things about our food supply chains and treatment of the planet in general. Parties that benefit from the status quo aren't really incentived to change that.

We are not shouting at the sky that much. All there is, is our satellite and terrestrial broadcasts. These are not intended to reach far beyond the earth and are encoded in a quite arbitrary way that is designed for easy implementation, not reverse-engineering. Interference and interstellar distances will likely distort them beyond recognition, unless you already know what you are looking for. There were some attempts at intentional broadcasting, but it's really not the main avenue of research (look for Active SETI). We are mostly listening at this point.

We are like an Uncontacted Tribe with technology. Even if we managed to decode a signal (not really feasible as many encoding schemes utilize quite arbitrary formulas and mapping tables), we would still not understand the language. Humans are pretty bad at figuring out languages without some sort of rosetta stone. Even if we understood their language, we might not be able to comprehend it, or comprehend it in fallacious ways, because of missing context. To compare, do our dramas, thrillers, TV shows, horror movies, porn (unfortunately often including children), and Youtube videos really portray an accurate picture of Human life and culture? We are ourselves quite bad at understanding other human cultures, past and present, and often rely on stereotypes...