The Earth has been broadcasting human generated radio signals for about 100 years now. If a nearby civilization were broadcasting similar radio signals, could we detect them with our own radio observatories like Arecibo or the VLA?
It's going to depend on the type of signal: early radio/TV v. the Arecibo message. Let's focus on the daily radio emission of modern life as opposed to the directed broadcasts that are designed to be observable.
Arecibo has a sensitivity (system equivalent flux density) of about 1 Jy for 1 GHz scale frequencies (Arecibo info). What is the flux density that we send into space? Is most of our stuff aimed at the ground and/or reflected back off the ionosphere?
I always bring this question up when I get talking about SETI, and I've never gotten a good answer. How far away from Earth could Earth be and still observe Earth?
At least one paper addresses this: Sullivan, et al.,"Eavesdropping: The Radio Signature of Earth". Science, 199:4327, 377-388 (1978).
I'm sure a lot has changed in our radio signature since 1978. Sullivan, et al. say:
The problem of detecting radio leakage from Earth as a whole is thus essentially identical to the problem of detecting its single strongest transmitter.
This is due to the non-overlap of signals (different frequencies by design plus differing doppler shifts for different regions). So the question can be rephrased:
What is the Earth's most powerful isotropic radio transmitter?