The answers in brief are: it is hard, but yes, almost certainly someone other than the DSN has done so.

So first of all how hard is it? The answer is it's pretty hard: you need very considerable resources to be able to hear signals from the current vehicles on Mars, and probably any past vehicles as well.

The hardware

A radio amateur, Scott Tilley (yes, that Scott Tilley: the same person who has found lost satellites) has heard signals from MRO and other spacecraft in transit to and orbiting Mars, using a dish of 0.6m diameter. So we know that these signals are detectable on Earth with fairly modest equipment by someone who knows what they are doing.

That's an impressive feat but not that surprising, because one of the reasons these spacecraft exist is to relay signals from landers to the Earth. Because they're in orbit they can afford to have large, fragile antennas: the high-gain antenna on MRO is 3m in diameter: that's about the same size as the whole of Perseverance which is 3m long. An antenna like that is not going to survive the EDL, and if it did it would entirely dominate the rover (where would the arm go?). That's why rovers don't carry such large antennas: they can't.

In contrast, Perseverance's high-gain X-band antenna is 0.3m in diameter, and its omnidirectional low-gain antenna is much smaller than that. The high-gain antenna hopes to achieve over 500 bits/second to a 34m DSN antenna on Earth and 3000 bits/second to a 70m DSN antenna. The low-gain antenna hopes to manage 10 bits/s to a 34m antenna on Earth and 30 bits/s (or even faster they say) to a 70m antenna on Earth.

There is also a UHF antenna which is mainly used to talk to the orbiters working as relays and which can do Mbits/s to them.

I'm not sure which system transmitted the tones that were used to indicate status during EDL: I think is almost certainly the low-gain X-band system, but it just possibly might be the UHF system. Presumably the high-gain X-band antenna was not deployed at that time, and even if it was it would not be possible to point it from a vehicle which is rotating and then swinging about under a parachute. These tones were, of course, detected on Earth by the DSN but there's almost no information in them.

Some mathematics

So, there are now two interesting questions: how does the signal strength on Earth from something transmitting from Mars go as a function of the various factors, and how does the ability to hear that go as a function of various factors (really: of receiving dish size). These turn out sort-of to be the same thing due to a wonderful set of theorems collectively called reciprocity.

Versions of the below formulae are fairly well-known I think. Enough information to derive them can be found on the Wikipedia page for parabolic antenna.

For transmitting, with a parabolic antenna on-axis, we have this:

$$ S = \frac{\pi}{4} \eta_t P_t\left(\frac{d}{\lambda r}\right)^2 $$

Where

• $S$ is received signal strength (power per unit area);
• $\eta_t$ is some measure of how efficient the transmitter is;
• $P_t$ is transmitter power;
• $d$ is the diameter of the transmitting dish;
• $\lambda$ is the wavelength;
• $r$ is the distance from the transmitter.

This ignores things like attenuation in the atmosphere, but we can ram that into $\eta_t$, or add some additional fudge factor. (I am in fact going to ignore the atmosphere in an attempt to make it easier to hear the signal.)

For the receiving system, assuming it's a parabolic dish we get this expression:

$$P_r = \frac{\pi}{4} \eta_r D^2 S$$

Where

• $S$ is signal strength;
• $D$ is antenna diameter;
• $\eta_r$ is the efficiency of the system.

So, putting this together we get this thing:

$$\tag{*} P_r = \eta_r \eta_t \frac{P_t}{16} \left(\frac{d D}{r\lambda}\right)^2 $$

In this expression the things we can usefully change are $d$ and $D$, and perhaps, at design time, $\lambda$, $\eta_t$, and $\eta_r$. But $d$ and $D$ are the interesting ones for us.

I have not addressed signal-to-noise here, and I'm not going to.

Some numbers

There are some numbers I know and some I don't know. I don't know $\eta_t$ or $\eta_r$ for anything: I will assume they are all the same (ie they're all pretty good designs, which seems plausible). I will also assume that everything works at the same value of $\lambda$: in other words I'll only consider the X-band system (this is much higher frequency than the UHF one used by the rovers to talk to the orbiters, and since $P_r$ depends on $1/\lambda^2$ it's therefore also much better). And I'll assume that $r$ is the same, which is true to a very good approximation at any moment in time.

As I said above I will assume there is no attenuation through the atmosphere of Mars, which is probably not the case, but ignoring it makes it easier to hear rovers on the surface.

One important number is the transmitter power. We know that MRO has a 100-watt X-band transmitter, and we also know that the total power available on Perseverance is 110W. I will assume that Perseverance can transmit at 33W: I suspect this is pretty optimistic and it's really 5-10W, but being optimistic makes it easier to hear it.

Finally I will assume that Scott Tilley was just able to hear MRO: in other words that a 0.6m antenna is what you need to hear it, and that the $P_r$ at which he can detect things is the same as anyone else's: in other words his receiver is as good as anyone's.

So we can rearrange (*) to get this

$$ \begin{aligned} P_t d^2 D^2 &= \frac{16 P_r r^2 \lambda^2}{\eta_r \eta_t}\\ &= K \end{aligned} $$

where $K$ is constant, since all its components are by assumption. So, knowing that Scott Tilley could just hear MRO we know that

$$K = 100 \times 3^2 \times 0.6^2\,\mathrm{Wm^4}$$

And thus we know that, to be able to just hear Perseverance on the surface

$$ \begin{aligned} D^2 &= \frac{100}{33}\times \left(\frac{3}{0.3}\right)^2 \times 0.6^2\,\mathrm{m^2}\\ D &= \sqrt{300}\times 0.6\,\mathrm{m} \end{aligned} $$

or

$$D\approx 10.4\,\mathrm{m}$$

And this is the number we're after: to be able to hear Perseverance on the surface you need a dish approximately 10.4m, or 34ft in diameter, you need a good X-band receiver strapped to the back of it, and you need to be pointing it at Mars.

This is well outside what amateurs have, to put it rather midly (perhaps if Bill Gates or Elon Musk or someone got interested in the subject they could afford such a thing). It's not that large by radio telescope standards, but almost everyone with a radio telescope to hand is using it for astronomy, not listening to spacecraft on Mars, so they both won't have an X-band receiver strapped to the back of it and won't be pointing it at Mars.

Who is listening

The DSN, obviously, have suitable facilities to listen to signals from vehicles on Mars. But we're looking for people who aren't the DSN.

Well, there are people with astronomical radio telescopes who will have an X-band receiver strapped to their telescopes and who are pointing it at Mars. For instance when InSight arrived at Mars, its EDL tones were observed by both Green Bank observatory in West Virginia and the max Planck's facility in Effelsberg, Germany. Although these people are not the DSN they perhaps still count as the DSN.

We can also assume that at least the Chinese have systems capable of hearing vehicles on the surface, as they are about to put one there. But, to my knowledge, they have not announced whether they have heard signals from the current vehicles on the surface: it's safe to assume they have however.

Who else is listening?

Is there anyone else with a suitably large dish, suitably motivated to listen? Well, yes it turns out there are: there are some people in Cornwall who have come into possession of some disused satellite communications equipment from a ground station formerly owned by the state telecommunications company. And they're in the process of repurposing this equipment, including their largest dish, 'Merlin', as a private DSN node, which will indeed have X-band equipment strapped to the back of it, and also possibly as a node in very large radio astronomy interferometer arrays.

These people are goonhilly.org (Cornish placenames are just wonderful), and the repurposed 'Merlin' is now GHY-6, and it's a 32m dish.

And on the 15th of February 2021 they made a press release which reads, in part:

Goonhilly has been working with the ESA over the past few weeks, using Mars Express - a spacecraft on an active Mars mission - as the test vehicle to validate their GHY-6 antenna. They are currently shadow tracking Mars Express as it orbits the red planet. Whilst shadow tracking Mars Express in this pre-operational phase, Goonhilly has also received signals from another global mission - the UAE Space Agency Hope Mission.

Picking up signals on February 9th, Goonhilly was able to report that the UAE Space Agency Hope Mission successfully entered Mars’ orbit. In a significant month for Mars missions Hope was followed by CNSA Mars Mission Tianwen-1 on February 10th. There are hopes that Tianwen-1, a rover-holding satellite, will touchdown on Mars in May.

Only a week after these two missions entered Mars' orbit, on February the 18th, NASA's JPL Perseverance rover is due to land on Mars's surface, and the GHY-6 antenna will be pointed at Mars during the landing attempt. Though not an official downlink, Goonhilly will be able to receive and potentially decode the signals from the rover should NASA need support before and during the landing. This will be the first time communications direct from the surface of Mars will be received in the UK.

[My emphasis]. Well, this press release was before the landing as you can see, and I can't tell you whether Goonhilly heard Perseverance on the surface of Mars, even if I knew. But unless anything went wrong they certainly did.

So, in summary: it is significantly hard to hear signals from vehicles on the surface of Mars, but people other than the DSN have almost certainly have done so: those people include, at least, Goonhilly.

I will update this answer when Goonhilly confirm that they did hear Perseverance on the surface of Mars.

Note that this answer originated as a draft for another question which was closed: I've edited it into better shape I hope but it may still retain some traces of its previous existence.