What do we do if SETI is successful?

For a somber, deeply intellectual view of what could happen, I can't recommend enough Stanislaw Lem's His Master's Voice: https://en.wikipedia.org/wiki/His_Master%27s_Voice_%28novel%...

"Given that our civilization is unable to assimilate well even those concepts that originate in human heads when they appear outside its main current, although the creators of those concepts are, after all, children of the same age—how could we have assumed that we would be capable of understanding a civilization totally unlike ours, if it addressed us across the cosmic gulf?"

Me and my dog cannot talk.

I understand my dog and he understands me.

If they experience death then we have massive common ground already.

I've never bought Wittgenstein's Lion for similar reasons. I am able to communicate with my cat, though it is not easy. We don't need language to do this.

It is also important to note that understanding is not equal. Certainly I understand my cat far better than she understands me. Famously Danish, Norwegian, and Swedish are mutually intelligible[0], yet this does not create equal understanding between all parties. Norwegians fair the best while Swedes are out of luck. It probably isn't surprising that this happens even when all speakers are speaking the same language. You can speak in front of 10 people and you may hear 15 different interpretations, none need be what you intended.

Language is messy. It's incredible communication happens with it. But we're smart creatures, and there's ways to establish frames of reference. We have theory of mind, even if we don't all use it. But using it certainly helps. Communication is best when all parties are trying their best to understand one another. Sometimes we confuse that to mean we're trying because we're talking. You're not trying unless you're considering what was intended to be said, despite the words used. To which, that, I agree is the lion.

[0] https://en.wikipedia.org/wiki/Comparison_of_Danish,_Norwegia...