Perhaps they've built their own comms system for example - maybe even a multi-site one that enables continuous contact - or maybe they're using NASA/ESA/JAXA assets. It would be interesting to know.
I'm not aware of any commercial providers for lunar communications.
They seem to either own or co-own all of the hardware
By this logic, there are huge amounts of land that I also “own” that guys with M16s (or, more accurately, M4s) will keep me from walking on if I try to go there. Half the year I live in southern Nevada, so this distinction has some direct practical consequences in my life.
It’s deceptive. Government property is not owned by citizens, it is under the exclusive control of the state—i.e. not you.
For example, if I own a water well, I don’t necessarily have the rights to do whatever I want with it. Some jurisdictions might let me pump out as much water as I want, but even those will punish me for blatantly polluting it (one would hope).
What some people think of as something akin to “total ownership” — completely unlimited access — would be tantamount to putting one’s “rights” above everyone else’s. Even dictators usually have some limits on their power, whether by laws, norms, or geopolitical pressures.
Across public and private spheres, the word “ownership” loses meaning. (Nobody “owns” NASA or the U.S. government, though they do “belong”to we the people.) That’s why, in ownership disputes between nations and under the law, the operant term is “control.”