It seems at least possible that Galadriel's knowledge that Frodo had put on the Ring was derived from tales told to her of the Fellowship's travels rather that from her ring. As someone else commented, if the bearer of an elven ring could perceive the One being put on, then Gandalf certainly would have had alarm bells going off when Bilbo put it on during the Quest of Erebor since he was wearing Narya.
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Beleriand, Beleriand,
the borders of the Elven-land.
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