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Originally this word had nothing to do with snooping.
Eavesdrop started off literally: first it referred to the water that fell from the eaves of a house, then it came to mean the ground where that water fell.
Eventually, eavesdropper described someone who stood within the eavesdrop of a house to overhear a conversation inside.
Over time, the word obtained its current meaning: "to listen secretly to what is said in private."