Fermi's paradox basically boils down to: if intelligent life is at all common, it should be everywhere we look, and yet we don't see any. The traditional position is that therefore if we don't see intelligent life, it must be because it's too rare, or communication across interstellar distances is impossible. But the other possible answer is that intelligent life is everywhere, and we just don't understand or even recognize it as such.
The fact that we've been living with dolphins for millions of years and actively studying them for decades and have only just worked out a pretty basic fact -- they have names for each other -- despite the fact that they share most of their genes with us and live under the same environmental conditions exactly, suggests that we are probably really really bad at recognizing intelligent life that isn't exactly like our own.
Radio waves are probably in the same bucket of technologies as smoke signals for civilizations few hundreds years more advanced than us.
For all we know we could be discussing how the transition to Kantian philosophy played out slightly different for them in a few years.
It was pretty cool. It gives me hope for a future of being part of a society with other sapient species in it.
But it wouldn't be quite the "whoa, on large enough scales the universe _really is_ the same, everywhere" type feeling that humans having meaningful conversations with, say, Fookin' Prawns would.
Given that the Milky Way alone has at least 100 billion planets (17 billion which are Earthlike)[1], it would actually be a far greater shock for there not to be intelligent life in space.
[1] http://www.popsci.com/science/article/2013-01/billions-and-b...
But it would be a complete mindfuck if we found other intelligent life in space, where before there were mainly only stars, rocks, and interesting formations of liquid and gas somewhere in between. It would change everything.