For the TL;DR, see the bottom of the answer. See also What would it take to get a message to another star? on Space Exploration SE.
We can approach answering this by considering how sensitive our most sensitive receivers on Earth are, how much antenna gain we can muster, how much power we can muster, and how much power needs to be transmitted for us to be able to detect the signal at interstellar distances. For simplicity, I'll just ignore the cosmic background radiation. In effect, this answer establishes an upper bound on how far away from us a civilization similar to us could be and we would be able to detect them.
The way to approach that is to construct a link budget for the transmission system and distance in question. The first thing we need for that is the equation for free space path loss, which is $$ 20 \times \log_{10}\left( 4 \pi \frac{d}{\lambda{}} \right) \approx -22 - 20 \times \log_{10}{\left(\frac{d}{\lambda}\right)} $$ where $d$ is the distance and $\lambda$ is the wavelength (where $\lambda = \frac{c}{f}$, where in turn $c$ is the speed of light in the medium and $f$ is the frequency). When $d$ and $\lambda$ are in the same units, the resultant value is the path loss in decibel (dB). Notice that the path loss scales with the distance in terms of wavelengths, so if you double the frequency (halve the wavelength) and halve the physical distance, the path loss is identical. On Earth there are other propagation modes as well (ionospheric, parasitic radiation, reflection, scatter, ...) that make calculating path loss far more complicated; however, for anything between different celestial bodies, this is the go-to equation for approximating path loss. It is only an approximation because it does not take into account for example losses in the interstellar medium.
For a reasonable frequency of interest, 3 GHz, the wavelength is 10 cm.
1 lightyear is, quite conveniently for our purposes, about $9.461 \times 10^{17} \approx 10^{18}$ cm.
$\frac{d}{\lambda} = \frac{10^{18}}{10} = 10^{17}$ so we calculate $-22 - 20 \times \log_{10}{\left(10^{17}\right)} = -22 - \left( 20 \times 17 \right) = -22 - 340 = -362$. Over 1 lightyear, our path loss is approximately 362 dB. For a more realistic example, let's take Proxima Centauri at about 4.25 lightyears from Earth; that gives us a total path loss of $-22 - 20 \times \log_{10}{\left(4.25 \times 10^{17} \right)} \approx 395$ dB.
(If desired, substitute the distance between the worlds of interest and an appropriate frequency in your case and recalculate the path loss. If you change the frequency, don't forget to recalculate the antenna gain below.)
Well, it's good that we now have a number, but what does that number mean?
The total worldwide electricity production in 2008 was 20,261 TWh, or if this were continuous (it probably wasn't) about 2.31 TW. Let's say we could somehow channel all of this into a transmission at 3 GHz (we almost certainly can't; even if we wanted to, there are efficiency limitations in real-world radio amplifiers, and with a really impure waveform, we might only be able to get maybe 90% efficiency in an amplifier which means we need to set our hair dryer to its 231 GW setting). The remaining about 2 TW is about +153 dBm. (If you compare this to the Arecibo observatory, you'll notice that the figure quoted by Wikipedia is an order of magnitude higher at 20 TW on the very similar frequency 2380 MHz. However, that's EIRP, which adjusts for antenna gain, while we are talking raw transmitter output power here. We'll get to antenna gain in a minute.)
A really good receiver, including a low-noise amplifier, might be able to detect a signal that measures something like -200 dBm at the antenna feedpoint terminals. The exact value varies with the receiver design and desired transmission rate, and I'm not completely sure what the state of the art actually is, but -200 dBm is likely close enough for our purposes particularly if the purpose is to simply detect the presence of the signal.
The really nice part about working with these numbers in dB relative to something (such as dBm, which is decibels relative to a milliwatt, or dB, which is just a ratio) is that we can simply add the numbers. If we feed that +153 dBm signal into the -200 dBm noise floor receiver, we have a margin of 353 dB for as long as we don't blow out the receiver circuitry (which would happen pretty quickly, but let's ignore that for a second).
The gain of a parabolic antenna (which Arecibo isn't, really; Arecibo is a spherical, not parabolic, reflector) is $$ G = \frac{4 \pi A f^2}{c^2} e_A $$ where $A$ is the reflector area in square meters, $f$ is the operating frequency in Hz, $c$ is the speed of light in m/s, $e_A$ is the aperture efficiency (defined as the ratio of effective aperture to physical aperture, or $\frac{A_e}{A_p}$), and $G$ comes out as a multiplication factor describing the antenna gain over an isotropic antenna (an antenna that has exactly the same sensitivity in all directions; no real-world antennas have such a radiation pattern, they are always somewhat directional). A good parabolic antenna might have an aperture efficiency in the 0.8 range, and I've seen (but can't seem to find again) a mention of 0.55 for poor ones. Because this is about interstellar communications, we'll use the best antenna we can reasonably muster, so let's call the aperture efficiency 0.8.
To give you an idea of its size, the Arecibo dish is 305 meters in diameter and weighs 900,000 kg. It's the largest single-aperture telescope in the world.
At 3 GHz, a 305 meter diameter parabolic dish with an aperture efficiency of 0.8 has a gain of $$ G = \frac{\left( 4 \pi \left( \pi \left( \frac{305}{2} \right) ^2 \right) \right) \times \left( 3 \times 10^9 \right) ^2}{\left( 3 \times 10^8 \right) ^2} \times 0.8 \approx 91811992 \approx 80 \text{ dB} $$ when compared to an isotropic antenna (EIRP gain). If we assume two Arecibo dishes pointed directly at each other, we can add another two times 80 dB of antenna gain to our link budget, so we get a bonus 160 dB for only a tiny difficulty in aiming (as anyone who has tried to aim a satellite dish can attest to; a household satellite dish has gain far lower than 80 dB at its frequency of interest). Here we can also see that even an antenna with a 1.0 (best theoretically possible) aperture efficiency wouldn't increase our gain by very much; such an upgrade would gain us another almost exactly 1 dB on either end.
So, to summarize, we are putting out +153 dBm, gain 80 dB in antenna gain, lose 395 dB along the way, gain another 80 dB in antenna gain, and need at least -200 dBm after all that for the signal to be detectable. Luckily, we are now at -82 dBm, so given these assumptions, the signal is well within the range of detectable. (In fact, I think my amateur radio transceiver could pick it up without much trouble, given an appropriate frequency downconverter. Of course, and perhaps thankfully for peace in the neighborhood, I don't have the Arecibo dish in my back yard.)
However, those are quite some assumptions that we are making in order to reach this conclusion. Basically, what we are doing is pouring the whole world's electricity production into the biggest radio antenna we can muster, expect the receiver to have an antenna just as large and that they are pointing it directly at us and are listening to just the right frequency at just the right time. Remove any one of these, and the signal goes from trivially detectable to anywhere between difficult and not a chance. The problem is exacerbated if we want to not just detect the presence of the signal, but also understand its content, at which point we start looking at noise over a larger frequency span and ultimately the Shannon-Hartley theorem, which gives a theoretical limit for the transmission rate of a communications channel of a given bandwidth and signal-to-noise ratio. Our terrestrial systems aren't really designed to be decoded at interstellar distances because, as pointed out in a comment to the question, companies like television networks and cell phone providers aren't really interested in Earth-bound investments in their Proxima Centauri viewership and customers.
For a real world comparison, SETI Sensitivity: Calibrating on a Wow! Signal from the SETI League indicates that the Wow! signal was received (in 1977, on 1420 MHz) on equipment that had a noise floor of -138.6 dBm (we are almost certainly doing better than this today) plus 55.3 dBi (dB over isotropic, or EIRP gain) antenna gain. Even if we would use such a receiver rather than our postulated -200 dBm receiver, but still use the Arecibo dish on 3 GHz, we "only" lose about 61 dB compared to the calculation above so we still have a margin of over 20 dB to the noise floor, which is quite decent and is going to stand out in any signal strength plot. (The Wow! signal peaked at about 30 times the ambient noise, equivalent to an about 15 dB margin. 20 dB means that the signal is 100 times the strength of the noise.)
As of September 2016, the Chinese are putting the finishing touches on what has been termed the five hundred meter aperture spherical radio telescope, or FAST for short, nicknamed Tianyan (天眼). While Arecibo has a 305 meter diameter spherical cap reflector, FAST has a 520 meter diameter spherical cap reflector (Five-Hundred Meter Aperture Spherical Radio Telescope (FAST) Cable-Suspended Robot Model and Comparison with the Arecibo Observatory, Ohio University, section 1) of which 300 meters is illuminated at any one time (The Five-Hundred-Meter Aperture Spherical Radio Telescope (FAST) Project, Rendong Nan et al, arXiv:1105.3794, doi:10.1142/S0218271811019335, PDF page 4 in the arXiv version). As such, it does not have significantly different properties as an antenna as compared to the Arecibo main reflector in situations where either can be used.
Scientific treatment of reception of incidental transmissions
(By "incidental", above, I am referring to those transmissions not actually aimed into space for the explicit purpose of being detected by an alien civilization.)
It appears that this has actually be discussed in proper scientific fora. For example, Rob Jeffries' answer on Astronomy SE on how we would detect an Earth doppelganger planet quotes Cullers et al. (2000) as stating that
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
and Tarter (2001) as stating that
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)
In other words, an alien transmission would need to be far stronger than a current, powerful Earth TV transmission, for us to be able to detect it with currently available equipment. There aren't a lot of transmissions that would meet this criteria.
TL;DR:
To detect the radio transmissions of a civilization outside of the solar system is absolutely possible, but realistically does take deliberate effort, helpfully at both ends. We won't be picking up anyone's cordless phone. Nor will we be picking up anything like our own cell phone networks, nor likely our Earth-bound point-to-point radio links. We might be able to detect the presence of an EM radiation spectrum that does not match either the cosmic background radiation or what you would expect from natural processes in a solar system. However, if they aim a powerful transmitter in our direction for some reason, and we happen to be listening at just the right moment on just the right frequency, then we probably would detect it.
