I can't answer your first question, but I'd guess that it isn't necessarily a function of physical size as much as signal-to-noise ratio, largely as a function of EIRP. Bigger antennas typically provide a higher gain factor (EIRP) but, unless they're connected to a very low noise receiver package, any improvement in signal-to-noise ratio is lost. Consider the antennas used in the Deep Space Network (DSN) to monitor things like the Voyager spacecraft. Voyager spacecraft are approaching one lightday from Earth and are radiating a few watts at most, but they can still be detected by the DSN. According to the DSN website, they've also developed the techniques to array several antennas to achieve much more sensitive signal detection.

The Galileo mission to Jupiter employed arraying in 1996-1997 to increase the science data return during its primary mission. For Galileo, the DSN arrayed up to five antennas from three tracking facilities (Goldstone, Canberra, Parkes) and two continents. The result was a factor of 3 improvement in data return compared with that of a single 70-meter (230-foot) antenna. Arraying, plus advances in data compression and encoding techniques, helped make Galileo so successful that its mission was extended to more than a dozen years of valuable observations of Jupiter and its moons.

Since the nearest star systems are several lightyears away, the signals being radiated would have to be considerable stronger than Voyager to be detectable by DSN type antennas. If some hypothetical alien civilization deliberately tried to signal us, they could radiate a tightly focused RF (or laser) beam of perhaps 100 KW (give or take a few orders of magnitude) towards the location where we'd be in the time for the signal to travel to us (a non-trivial aiming problem, when you think about it).

As you asked, why would they do so? About the only reason that comes to mind is that they were able to detect our RF leakage and were trying to signal us. That makes the assumption that they have the technology to detect us while at the same time, some are claiming we couldn't possibly do the same in the other direction.