Here’s the setup…say there’s an Earth-like planet orbiting Alpha Centauri A. Not only does it have life, and intelligent life at that, but by an amazing coincidence, for the last 150 years or so, their world’s technological development has kept pace with and equaled (not surpassed) Earth’s.
My question is: how soon, likely, could each planet have noticed the other’s existence—I’m assuming a few years of constant TV and radio noise would get noticed somehow—and how difficult would it be to establish any kind of communication? (I mean, I know we wouldn’t be getting realtime two-way video, but how much effort would it take before we could at least start regularly “shouting” binary numbers at each other?)
The Alpha Centauri stellar system is only 4.35 light years away from Earth. If they have been lucky enough to be able to pick up our broadcasts over the natural background noise of interstellar space, they might have discovered our existence in the 1930s or 1940s when strong radio broadcasts became commonplace.
If they are of our intelligence and technological level, they’ll know about the prime numbers, the Fibonacci sequence, the perfect squares, and other essential integer sequences. So broadcasting one of those sequences (I think the prime numbers are favored) on a clear wavelength (the resonant frequency of hydrogen, I think, is favored) would be a very clear indication to them that Someone is Over Here. Of course, that assumes they’re listening. Maybe the answer to the Fermi Paradox* is that all of the aliens are massive MMORPG freaks and don’t care about interstellar life. 
*(If they’re out there, why don’t they ever call?)
Once they know where we are, it shouldn’t be difficult for them to broadcast something on the same frequency they heard us on. They’ll probably pick the primes as well, simply to acknowledge that they heard us and are going to try to respond. From there, who knows? That would count as First Contact, and we have no experience with that at all. Will we be able to understand them? Will their perceptions of the Universe mesh with ours in any useful manner? I think they will, simply because we share the same laws of physics, but the conversation will probably be extremely halting and difficult.
It is possible to write a self-unpacking message that begins with integer mathematics and works its way all the way up to biochemistry and descriptions of the planet we live on.
Here is one such message; all that is required to understand it is:
-a grasp of the concept of persistent symbols (this might be the biggest and least safe assumption of all)
-a basic knowledge of mathematics, including prime numbers and an awareness of different number bases
-a basic knowledge of chemistry (chiefly atomic weights etc) and physics.
A couple of years back, I downloaded the message pages without any of the accompanying explanation and was able to decode most of the message without knowing anything about it at first.
4.35 light years is close enough that actual conversational communication could occur within the lifetime of individuals (at our end, at least - probably shouldn’t assume anything about lifespans, or indeed individuals at the other end).
The ping times would be murder though.