How could an amateur astronomer who lived her whole life on the far side of the Moon verify that Earth existed?
Well, not by tuning in episodes of reality TV. Since the Moon has virtually no atmosphere, there is no mechanism (like ionospheric skip) by which radio signals from Earth or its communications satellites could propagate around the Moon’s curved surface and reach a point perpetually facing away from Earth. Even when we on the home planet see a total lunar eclipse, there would be no visible sign to an observer on the Moon’s far side that Earth was blocking out the Sun’s light.
Earth would betray itself indirectly, however. Our far-out astronomer would have a much harder time than Ptolemy or Copernicus puzzling out the motions of the solar system. Superposed on each planet’s direct motion and graceful retrograde loops — the same ones seen from Earth — would be a mysterious monthly deviation. Mars, for example, would advance and retreat along its smooth track by up to 1/3° each month, owing to parallax as the Moon revolved around the Earth-Moon barycenter.
— Roger W. Sinnott