The original of this GIF was tweeted here and I've reconstructed it because I don't know how to post it here (How to show this particular tweeted (GIF?) in a question? - trouble downloading). You can see it is possible to pick up signals even from cubesats at the Moon with a 25m dish (hat tip to @Hobbes), so receiving data from a well-powered and optimized relay satellite like Queqiao would need only a more modest dish.
On 7 October, the Chinese lunar satellite DSLWP-B made this 1.5 hour time lapse of the Earth appearing behind the Moon. A team worked hard this week to download the images with the Dwingeloo @radiotelescoop: @bg2bhc, @KuehnReinhard, @ea4gpz, @tammojan, @cgbassa, PE1NUT et al
You can read more about this dish in
China has a satellite communications station in Argentina, which may be useful, and as @GremlinWranger points out tracking ships such as the Yuan Wang-class tracking ships few shown below) if it really needed to extend its geographic stretch. Yuan Wang 2 is shown docked in New Zealand for example.
While someone is going to say "but the ships are for tracking launches" I'll just say that the ships are for holding antennas, and they can be used to listen to anything that the government would like to use them to listen to.
From China deploys Yuanwang-7 space tracking ship ahead of Tianzhou-1 cargo mission:
Yuan Wang 7, constructed by Jiangnan Shipbuilding, is 220 meters in length, 40 meters tall, displaces 25,000 tons, and is said to be able to can resist strong typhoons while operating 100 days at sea, the department said in a statement.
The ship is said to be the most technologically-advanced watercraft designed by China for the use of space tracking. The vessel features three large dish antennae some 10-12 meters in diameter, and an array of radomes and aerials.
The Yuan Wang-class ships supplement China's network of land-based tracking stations, which span the breadth of the country's 4,500 km-wide land mass.
China built its first monitoring ship, the Yuan Wang 1, four decades ago, becoming the 4th nation on the planet to deploy this type of vessel, after the US, Russia and France. Since then, the Yuan Wang fleet has carried out some 70 expeditions, traveling over 1.5 million nautical miles in the Pacific, Atlantic and Indian oceans. Examples:
This video Chinese space tracking ship completes maritime monitoring mission for Tianzhou-1 explains (in English) that the ship tracked the spacecraft in orbit and all the way until Shenzhou-11 docking TianGong-2 space station. So it really is used as a portable "ground" station for communication with spacecraft in Earth orbit, and since Queqiao is really in an Earth orbit (it's not really in orbit around the Moon), it could certainly be used in this case as well.
All-in-all, there's no reason to expect that communications is limited to the Moon being visible from China's landmass.
Amateur radio operators listening to Apollo transmissions from the Moon with a modest size dish antenna and 1970's electronics. You don't need a big dish to communicate over 0.0027 AU.
From Tracking Apollo-17 from Florida, borrowed from the question Why such a large observed doppler shift from Apollo 17 orbiting the moon?
This tweet is linked in the article A tiny Chinese satellite is orbiting the Moon and allowing radio amateurs to download images and shows an even smaller antenna can be used to at least pick up a carrier.
Borrowed from Help understanding this tweet about receiving images from satellite behind the Moon.
Borrowed from What will China's new “space station” in Argentinian Patagonia be used for?
The New York Times (fairly strongly worded) article From a Space Station in Argentina, China Expands Its Reach in Latin America describes a "space station" or base equipped with (among other things) a dish antenna to receive signals from space. The article is long, and well worth a read. Photography is beautiful as well.
Many more images in @Rob's answer!