The receiver would be much, much bigger than a square kilometer by many (>6) orders of magnitude. Any civilization capable of interstellar travel would be very capable of setting up a planetary or solar system sized astronomical interferometer (aka hypertelescope [1]) using a large fleet of synchronized satellites. These satellites can be prototyped with today's technology and any interstellar civilization would be capable of both receivers and transmitters with far more power than we have attempted today.
That's if you're trying to reconstruct a "picture" of what the interferometer is pointed at. If you control both the transmitter and receiver, you can use many different techniques to pack more information into the same amount of power. 2G cellular radios, for example, were 2 watts max power while 3G and 4G radios are 500 milliwatts max power despite being orders of magnitude faster thanks to better encoding due to radio sensitivity and spatial resolution due to MIMO - the cellular equivalent of an astronomical interferometer.
The technical term is space-time block coding [1]. The speed negotiation would take at least 3x the distance between the two start systems though, until which time they'd be limited to a conservative estimate based on the channel impairment between the star systems, aka max expected noise between the two astronomical phased arrays.
[1] https://en.wikipedia.org/wiki/Astronomical_interferometer