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?
Update:
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?
Best Answer
Using current technology (and by that I mean experiments and telescopes that are available now) we would probably be unable to detect radio signals from Earth even if observed from a distance of a few light years. Therefore there is currently no prospect of detecting such signals from (around) another star.
If we are talking about detecting "Earth", and assuming that we are not talking about deliberate beamed attempts at communication, then we must rely on detecting random radio "chatter" and accidental signals generated by our civilisation.
The SETI Phoenix project was the most advanced search for radio signals from other intelligent life. Quoting from Cullers et al. (2000): "Typical signals, as opposed to our strongest signals fall below the detection threshold of most surveys, even if the signal were to originate from the nearest star". Quoting from Tarter (2001): "At current levels of sensitivity, targeted microwave searches could detect the equivalent power of strong TV transmitters at a distance of 1 light year (within which there are no other stars)...". The equivocation in these statements is due to the fact that we do emit stronger beamed signals in certain well-defined directions, for example to conduct metrology in the solar system using radar. Such signals have been calculated to be observable over a thousand light years or more. But these signals are brief, beamed into an extremely narrow angle and unlikely to be repeated. You would have to be very lucky to be observing in the right direction at the right time if you were performing targeted searches.
Hence my assertion that with current methods and telescopes there is not much chance of success. But of course technology advances and in the next 10-20 years there may be better opportunities.
I did read (struggling to locate it now) that SETI were targeting multiple transiting exoplanets. The idea here is that you can wait for the two planets in the transiting planets to come into line and then hope that some beamed signal from the inner planet to the outer planet "spills over" and heads towards the Earth - ingenious.
It has been suggested that new radio telescope projects and technology like the Square Kilometre Array may be capable of serendipitously detecting radio "chatter" out to distances of 50 pc ($\sim 150$ light years) - see Loeb & Zaldarriaga (2007). This array, due to begin full operation some time after 2025 could also monitor a multitude of directions at once for beamed signals. A good overview of what might be possible in the near future is given by Tarter et al. (2009).
Loeb & Zaldarriga give a table (Table 1) in their paper that lists some sources of transmission from the Earth (which are the basis for their claim). The most powerful pseudo-isotropic sigmals appear to come in the 40-850 MHz range from TV transmitters, with a summed power of $10^{9}$ W or $10^{10} $W/Hz.