Suppose money and engineering wasn't a concern. Could we actually build a bunch of radio telescopes in space, and use radio interferometry, so that we could actually hear the sort of radio transmissions regular humans put up in space ?

I know radio transmissions become indistinguishable from cosmic background noise after a long enough distance. I'm assuming this would also apply to my imaginary telescope. Is there a maximum size the telescope swarm can be, after which increasing the size further brings no benefit ?

For the sake of discussion, let's assume there is a signal out there that we might possibly detect.

This question hits on one of the big weaknesses of interferometry as compared to a really huge telescope. Using two or more smaller telescopes, you can use interferometric techniques to produce an image that's as clear as what the bigger telescope could make, but the downside is that it's much "dimmer" than the big scope's image. You can only pull in the number of photons that are actually incident on your detectors. So when you have a relatively bright object, interferometry can get you a more detailed image, but it's not great for looking at very dim objects.

Signals intended for even interplanetary use within a system are going to be extremely weak by the time they reach another star, so you're going to need to catch as many photons as possible to distinguish the signal from the background noise of the universe. In this context, I don't think interferometry actually does anything to help -- you don't need a more detailed image, you need a strong detection, and that means you need a large receiver area, which suggests you need a really big dish.

And sure, you could potentially construct a vast swarm of smaller satellites to get you the same detection area as a huge dish, but at that point it seems like you're moving into an area where the distinction between a telescope swarm and a really huge segmented reflector starts to break down.

Radio waves are a part of the electromagnetic spectrum, and if transmitted using an omnidirectional antenna, they are subject to the inverse square law, which shows how the initial signal strength (or light intensity, if the source is a point of light) decreases with distance.

To get an exact answer, it is necessary to know (1) if the signal is directional (such as sent via a parabolic dish in a certain direction, in which case it will attenuate at a lower rate) or omnidirectional; and (2) the initial signal strength. Regular radio transmissions are omnidirectional.

See also this question which mentions Voyager, the most distant probe sent by NASA, and antenna required to detect the signal, though note that the Voyager probes use a directional dish antenna to transmit the signal, and the Voyager 1 signal requires 22 hours and 35 minutes to reach Earth at this time.

The Deep Space Network dish antennas receiving the signal are 70 meters in diameter, and the dish antenna on the spacecraft is 3.7 meters in diameter.

The largest radio telescope is 500 meters in diameter.

Regarding the signal strength issue, see this question which points out that radio signals are also photons, just like those emitted by stars. On the night sky only stars that are a few hundred light years distant can be seen easily - although each star is a far more powerful electromagnetic transmitter than an artificial transmitter.

It may also be worth considering that radio transmissions such as those sent out in space from Earth, are likely sent only for a certain time period in the development of a human civilization, probably a few centuries, until a more appropriate technology is discovered, because radio signals, lasers, and the electromagnetic spectrum in general, is not useful for communication at interstellar distances.

To illustrate, the closest star system to our sun is Alpha Centauri, at 4.3 light years distance - so if an email is sent there by radio signal, the reply would arrive at the earliest after 8.6 years (round trip time).

Our galaxy is 120,000 light years in diameter, so any electromagnetic signal received from a distant planet, would be from its distant past as well - and if the window of time where the people on that planet used primitive radio signal technology - has already passed the Earth, like a wave crest, and they are now using something more advanced which we do not detect, then we hear nothing, even if a large enough radio telescope dish would exist.

So a dish would have to be very large to receive a radio signal only from the nearest stars, and would have to be located in space, shielded from the sun by a planet, or even outside our solar system, and even so, this would work only if the humans living on planets around nearby stars have not yet passed the radio signal stage in their technological development.

This would be like a civilization trying to contact us using smoke signals, or looking for our smoke signals, and receiving no reply/sign of life, because we use radio signals, not smoke signals.