First I'd like to talk a bit about statistics. Our planet has existed for some 4 billion years-ish, and support human-like life for perhaps half that, if im being generous.
Of those two billion years, humanity has been intelligent enough to looks at the stars in a scholarly fashion for perhaps 3000, again if i'm really generous. Lets say it takes us another 1000 to set up a bus to go to the moon and back.
That's a time frame of 2 000 000 000 years that a planet has stupid life, compared to 4 000 years of intelligent life. Meanwhile, with such a fertile planet that close, humanity would spread across it's surface like mice in a barn. If the moon were fertile and of even remotely suitable atmosphere, you can be pretty sure we'd have a colony there already. Statistically speaking, either intelligent species will colonize the other planet long before a species can evolve there to be about as smart.
To make your story believable, I hope you got something to remedy these statistics.
As for communication, I think radio is the first method available. With a good telescope, you can figure out there's life on the other rock a couple hundred years early, basically as soon as you get higher quality glass working for lenses. 335
However, how would you agree to any communication protocol ? You probably can't. Every communication protocol we use, both sides have an understanding of how it works, we encode and decode the information. This was a consideration for the old voyager drones - part of what was done was engraving pictures of humans in a plate of gold, as well as an attempt to codify a small message in a sort of language we hope is easy to understand, but all our perspectives are human, not alien. Your races would have likewise problems.
It's not like talking to other humans. It's like talking to animals. With a lot of training, we can for example teach sign language to certain types of monkeys. However, establishing any mutual understanding that way requires close contact. Even if you found geeks on either planet who talk by radio and attempt to make sense of what the other one's trying to say, chances they'll reach some understanding in any short amount of time is really slim. 209