I've just read North Korea’s “successful” satellite in orbit, but tumbling and useless. How can we tell a satellite is tumbling or not? It's so far away!
First, let's clear the part about how far away Kwangmyongsong 4 (KMS-4, NORAD ID 41332) is. It was inserted into a 472.6 km × 508.5 km near-polar sun-synchronous orbit with inclination of 97.5°. It is not geosynchronous, so its orbit takes it over all the longitudes of the globe, and latitudes of 80.5° North to 80.5° South. That means that roughly once per day, it will be nearly directly overhead over any spot of the Earth's surface, at an altitude of 472.6 - 508.5 km above mean sea-level. You can see its orbit for yourself with tools such as Satflare, N2YO, Heavens Above, and so on.
It's orbital period is 94.3 minutes, and its nodal precession rate is approximately 1° (360/365.25) per day. That means that maximum equatorial separation between its ground tracks is 23.5° or 2,616 km. Half of that is our worst case observational angle to it relative to zenith from any point on the Earth's surface at least once per day, as long as it's within the ±80.5° latitude. That means that we can observe its passes with fixed ground facilities every day at a distance of between 472.6 and 1,405 km and at higher latitudes, this ground track separation decreases further still.
That's not too bad, and we can track so distant objects down to about a couple of centimeters in size even with civilian ground based radars, such as e.g. NASA's Haystack Auxiliary Radar, and who knows what capability military has, but I wouldn't put it past them that they use active radar and are illuminating the object with phased array microwave systems to get even better resolution and increase the object's radar signature. In fact, NASA's Deep Space Network is capable of achieving a similar feat on their own, and that's not a classified system.
OK, so how do we know it's tumbling out of control? We can simply infer that via periodically changing radar signature. A tumbling object that isn't spherical and reflecting in observational wavelengths uniformly in all directions, will produce sinusoidal dips and increases in brightness of the reflected photons (in whichever wavelength). If that spin rate is too fast to be deliberate, you can reasonably conclude that it is spinning out of control and it isn't simply a case of a curious attitude control. The object is roughly 200 kg in mass, and there is a limit to its ability to neutralize its angular momentum.
Also, if the intelligence community leaks such information despite the fact that it would suit them better that the object remains a threat to someone's national security and they get some nice new toys, erm, assets and/or permission to scratch an itch, pardon, to mitigate the threat, I would tend to believe them.
And if you prefer to believe the civilian society, there's even no need to believe anyone focused on their own national interests. Remember Progress M-27M? While we knew it was tumbling out of control because we saw the start of that in a live webcast from its own camera, and there was no reason to doubt official press releases, even amateurs and all over the world were able to confirm its spin rate. While it would take a bit better equipment to do the same for around 40 times less massive object at twice the orbital altitude, I wouldn't put it past the more persistent enthusiasts to confirm this for KMS-4 as well. Stay tuned.