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I'm sorry to say that putting a coil near a tower would only leech what passed through it, so unless the coil was nearly the same size as/bigger than the tower, and right beside it, and the coil was directly in line with the folks reporting bad reception, this is probably a tall tale.

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Not true, the coil emits its own radio waves which are exactly 180 degrees phase shifted with respect to the radio tower. Destructive interference could be measured behind the coil, even when not exactly in its shadow.

Wouldn't the coil emit the opposite induced field?

Yes, definitely, but only to the degree that you actually consume it's generated power.

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