If I can trust arbitrary estimates on stack exchange, the deep space network transmitters will be about equal brightness to the sun at frequencies near 2GHz when the 20signal is 10 Hertz wide. That would put the transmission rate in the ballpark of 10 bits per second, and the real number is 16 bits per second, so that seems to work out.

Being drowned out is harder than you might think. The maximum data rate of a weak signal is 1.4 x [bandwidth] x [signal-to-noise ratio]. If you transmit across a 200MHz band, and your signal is a million times weaker than the noise, you can do hundreds of bits per second.