But it's not a paradox at all.
Imagine there's an alien civilization in our galactic neighbor Proxima Centauri. How would we know they are there?
To make it more plausible. Let's say we figure out a way to send a space probe to Proxima Centauri. How do we communicate with that probe? Easy, you say, just like we communicate with the Voyagers: they point a reasonably large parabolic antenna at us, we point a point a much larger one at them, and we chat all day long.
Except that Proxima Centauri is 2000 times farther that Voyager. And signals decrease with the square of the distance. By the time Voyager gets to be 4.25 light years away from Earth, their puny 22W antenna will send us a signal that's 4 million times weaker than what they send us now (when we get less than 200 bits/second).
It's actually worse than that, of course. If Voyager were to send a signal from some arbitrary place 4.25 light years away, we'd get it as something 4 million times weaker. But if that place is in the solar system of Proxima Centauri, it's like us looking straight at the headlights of an incoming car. That very weak signal will need to be extracted from the extraordinarily bright noise coming from the star that's roughly speaking about 1 second of arc away from it.
Now imagine that instead of Voyager, it's just some aliens that are not pointing an antenna straight at us, but just minding their own business. They could be launching rockets, watching satellite TV, waging wars, and from time to time lobbing Himars at each others, and even nukes. How would we know they are there?
And now finally imagine that there are no aliens in the Proxima Centauri system, by the luck of the draw, but there are in 5% of all the star systems in the galaxy. Only the closest ones to us are 40 light years away, instead of 4. That's another reduction in signal by a factor of 100.
The only question is not why we didn't "see" any aliens so far. It's why we wasted so much money on SETI. Was Carl Sagan not able to do the back of the envelope calculation I just did here?