They chose Aricebo because it allows them to use very low power antennas to communicate. The size of the dish makes up for the lower power transmission.
In the past, they used high-powered antenna arrays.
For short pulses momentary power output reaches Terawatts!
They've said in the past (I'm failing to find the reference right now) that they chose Arecibo because of the high-gain of the antenna. I believe that the amplifiers that they're actually using above the dish are somewhere around 700 Watts.
Antennas with large effective apertures (of which Arecibo's is the largest) are considered high gain antennas, which have small angular beam widths. Friis' Transmission Equation shows that you could use low-power transmitters, but still maintain high boresight with a sufficiently large antenna surface. :)
http://spacecollege.org/isee3/isee-3-reboot-amplifier-instal...
It is an AR 700S1G4 with rated minimum output power of 700W, and will be used in S-band. Here's the datasheet:
http://www.arww-rfmicro.com/post/700S1G4.pdf
At these frequencies, the Arecibo dish has a gain of around 72dBi, putting the EIRP at about 7 gigawatts.
For scale, a typical wifi router has an EIRP of around 50 milliwatts, so this is about 140 billion times more powerful.