I confess, I was kind of baiting with the "decode the transmission" part. As someone else mentioned, the way we send the info doesn't have to be as complicated as what you're thinking. Many books and movies have been made about the subject... Contact and Project Hail Mary come to mind immediately. SETI is dedicated to being on the receiving end and makes assumptions about what ET would be sending based on assumptions about what we would send (filtering for repeating patterns and mathematical structures on narrow frequency bands).

In any case, how we transmit and what we transmit are two different things. What we transmit will likely be math and/or cosmos related, things we have in common with our galactic cousins (we assume). How we transmit... There are a million ways, so we narrow it down to what we think is the most likely to reach the target and be something they're listening for. And cross our fingers.