It is a quite elegant and simple experiment for any Flat Earthers who wish to deduce for themselves the true distance of the Moon.
No, you cannot determine this for yourself. The process involves sending your device information over the internet to get a response from a large facility.
See this video of a Moonbounce:
https://www.youtube.com/watch?v=TUbNDNBPFEMHe is communicating with a third party installation, and says "he heard me" and at 4:54 that "you need a pretty big station at the other end".
The Moonbounce is not conducted by setting up your own antenna, broadcasting a signal, and then receiving it. The process involves sending your data over the internet to a large (likely government funded) radio astronomy facility and then receiving back the results.
This "evidence" essentially involves asking the government how far away the Moon is. For what reason this service was made open to the public is unknown. But we may as well just go to the NASA website if we are relying on the government for our information.
From an article on the Moonbounce:
https://www.electronics-notes.com/articles/ham_radio/amateur-propagation/moonbounce-propagation-eme.phpMoonbounce basics
The basis of operation of Moonbounce or EME, Earth-Moon-Earth is the use of the Moon as a passive reflector. In view of the very large distances involved and the fact that the Moon's surface is a poor reflector the path losses are colossal, but nevertheless it is still a form of communication that is theoretically possible to use, and one that many radio amateurs regularly use.
There are clearly two stations in this diagram, not one. References can be found that the technique was developed by the U.S. Military after WWII.
From another Moonbounce reference of someone in Antarctica making a Moonbounce:
https://search.informit.org/doi/pdf/10.3316/informit.535111485198853Amateur radio operator Craig Hayhow has used the moon to bounce
a radio signal 742 000 km, from Mawson station in Antarctica to
Cornwall in England.
The Cornwall facility mentioned here is likely the Goonhilly Earth Station and 32-Meter Dish used for Moonbounces operated by the ESA:
http://www.arrl.org/news/goonhilly-32-meter-dish-to-be-active-on-moonbounce-on-september-1-2Goonhilly 32-Meter Dish to be Active on Moonbounce on September 1 – 2
A team of moonbounce enthusiasts expect to activate the 32-meter antenna GHY-6 at Goonhilly, on the Lizard Peninsular in Cornwall (IO70jb) in the UK on September 1 – 2, operating as GB6GHY. The group, including G8GTZ, G8GKQ, and G4NNS, will be on the HB9Q logger while operational, which should be between 0800 and 1200 UTC, but “earlier if possible,” they’ve said.
GB6GHY will concentrate on 3.4 GHz on September 1 and 5.7 GHz on September 2, with the ability to switch bands immediately.
“Anyone with a relatively small dish (3-meter or less) should be able to work us,” their announcement said. The European Space Agency is undertaking a project to upgrade Goonhilly Earth Station to track missions to the Moon and Mars. The work will see the GHY-6 antenna — which carried the 1985 Live Aid concert around the world — upgraded over the span of 2 years.”
This cunning proof is a service that a space agency provides.
The assertion that this is a proof that amateur operators can perform for themselves is incorrect.