The Fermi Paradox

As an example, the European Conquest of the Americas was driven by competition between European nation-states, it might not have occurred so swiftly or at all without some kind of competitive pressure.

Said another way, imagine yourself waking up in the court of King James of England or Queen Isabella of Spain. You establish rapport and they believe you come from the future. You communicate faithfully what you know about history. What conclusion do you think they are more likely to reach?

1) They have done the right thing by colonizing the Americas.

2) They have created what will eventually become new competitors, of whom at least one (the USA) will have subjugated them under the disguise of an alliance called NATO.

In the crucible of 15th-century European politics, neither England nor Spain may have had much choice in their desire to acquire more resources and open new markets.

And maybe the 21st Century will have cheap space travel and good reasons to take the trip. But we shouldn't ignore the possibility that Earth's governments may just decide to not let anyone leave and blast anyone who tries. Let's call it the "Nitrogen Curtain" in honor of Winston Churchill.

We can infer life based on atmospheric compositions NOW, but only if that planet fits an extremely narrow margin... meaning, Earth. Earth three and a half billion years ago had life on it, but you'd be hard-pressed to tell it from orbit.

If you want to move to unambiguous (intelligent) life then.......

The largest radio telescope we have (the 305 meter diameter Arecibo) would need to have it's sensitivity increased by around two orders of magnitude JUST to pick up our TV/FM/AM signals from outside the solar system. So, the crap we pump out the most couldn't even be detected by Voyager 2, if we had strapped a giant dish to it.

If we move into the narrowband signals then, depending on the source-strength, Arecibo could potentially pick up signals at up to a few thousand light years... if it happened to be pointed in exactly the right direction at exactly the right time.

So, our most sensitive instrument is only capable of measuring a fraction of a percent of the spectrum in a fraction of a percent of the possible-directions-it-could-be-pointed if the point source happened to be sending a strong enough signal (aimed at us) at exactly the right time (which would be anywhere between 2 and a few thousand years ago).

I haven't done the math, but I suspect that regardless of the frequency and amount of energy we dump into sending out a signal (within the realms of not being scifi, anyhow), it would be impossible for anyone to detect us a thousand or two lightyears out (using EM radiation) without knowing when/where to look. They also wouldn't be able to "see" that for quite a long time yet. I couldn't find a list of "when did we start sending out signals at X frequency" (and I suspect it doesn't exist), but if we take 50 years ago as a guess and we assume we pumped out noise at a frequency that could make the distance at a power level sufficient to be noticeable and that there wasn't anything to get in the way of the signal then we're only looking at something like 2000 star-systems.

The largest radio telescope we have (the 305 meter diameter Arecibo) would need to have it's sensitivity increased by around two orders of magnitude JUST to pick up our TV/FM/AM signals from outside the solar system. If we move into the narrowband signals then, depending on the source-strength, it could pick up signals at up to a few thousand light years... if it happened to be pointed in exactly the right direction at exactly the right time.

You'd have to build an absolutely monstrous dish to be able to detect any of our signals at even the center of our own galaxy (much less at the other end)... I haven't run the numbers on it, but I doubt it's physically possible to build a receiver large enough. Add to that that they're highly directional devices and I think you'll find the "N" in your scenario is actually a startlingly small number.