As mentioned before, this is untrue. In radio frequencies, the Earth is far brighter than the Sun. A civilization capable of radio astronomy within, say, 80 light years of Earth would have little trouble picking up our radio signals. Whether they would conclude the signals were from an intelligent civilization is another question entirely. They would, however, surely be detectable.
Signals we sent out 80 years ago (1929) would just now be reaching a star 80 light years away. According to
this source, broadband broadcasts (like radio and TV) would be undetectable beyond our own solar system even with a radio telescope 100 times the sensitivity of Arecibo.
ETA: And, as previously pointed out, MOST of the stars in the galaxy lie much further away than 80 light years. We've only been broadcasting radio waves for about 100 years. The claim was that beings "a great distance" from us could easily detect life is false. Most stars "a great distance" from us haven't even received light from our sun emitted when humans existed. Sure you could assume some magic technology that allows them to grab those signals long before they reach them, but that's not what I would call "with comparative ease".
I doubt such a civilization exists, but it is chauvinistic to assume that just because we can't detect Earthlike planets at interstellar distances, E.T. couldn't do it.
It is equally fallacious to assume that something which is impossible to us is not only possible but "easy" for an ET intelligence. Especially if this is then used in an argument that says we are unique in the galaxy. (The standard is intelligence like us, not intelligence far in advance of us. Argument based on the absence of evidence is weak enough, but this is moving the goalposts.)
ETA: This is the same problem I have with using the Fermi Paradox as "proof" that ET intelligence doesn't exist. It assumes the existence of technology that is impossible by current understand. I'm not asserting that means it is certainly impossible, but it's a weak argument that assume something like that. (And assumes that if it's possible it will happen--as if there were no other barriers to achieving anything that is possible, such as economic limitations, civilizations not enduring long enough to achieve all that is possible technologically, lack of will to do such a thing, etc.)
For the record, I believe that intelligent extraterrestrial life does exist in the Milky Way: the statistical argument in favor of it (to me) is quite compelling. The problem of detection, however, may well be insurmountable (at least in our lifetimes).
I don't know whether it exists or not. Assuming that we are somehow special or unique hasn't panned out in the past, and there's no reason to think that something like what happened here can't happen elsewhere.
I agree with you that even if intelligent life is relatively "common" we are still not likely ever to encounter it because things are so spread out in space and time. (My issue with the term "rare" is that it is a relative term. That's why I'm focusing on amb's assertion in absolute terms: that we are unique in the galaxy and there are probably no more than 12 such intelligences in the entire universe.)