In my story, humanity has done exactly that. In fact, Dark-Side Station is one of a couple of human colonies that have been established on the moon. While most people prefer to be on the Earth side of the moon with its magnificent sights of our home planet, there are some who prefer the purity of space. One character (whom I have not yet fleshed-out) has specifically started using the radio-telescopes to search for broadcasts from alien civilizations.
Despite high hopes for the project, he (or she) is still quiet surprised to pick something up. The signals are garbled and confused, but she (or he) is eventually able to figure out a way to clean up and translate the information, discovering that they are (basically) radio and television signals. The aliens are similar to us in many ways, but clearly alien. Linguistic experts pore over the data and eventually figure out the languages used. People on Earth start listening to and watching the translated broadcasts not just on a scholarly level, but for pure entertainment.
Here is where I can branch in several directions:
1. We manage to establish communication with the aliens and start a long distance dialogue despite the massive time-lag (we're talking decades) between communications.
2. Wars erupt on Earth over the alien contact with many people claiming that the signals are governmental forgeries designed to disrupt world religions (I've already seen sites that claim this idea, by the way).
3. We watch, helplessly, as the alien civilization destroys itself.
4. We realize that the transmission is from centuries, if not eons ago.
5. We watch as the alien civilization comes into contact with our media and decides that we are a hostile species. They then create a space fleet in order to gain a first-strike advantage.
The first one was my original thought, but there's no reason that aspects of all five might not be mixed together.
Whatever direction I decide to go, I certainly hope that our government (or someone) decides to build Dark-Side Station in reality. Someone else out there is bound to have started shooting messages into the stars.