I seem to recall that in the books, Gollum said "Shire...Baggins" as well. And the riders, after seeing that Frodo was not in the Shire (nor the ring...nor Bilbo for that matter) probably tried to follow the power of the ring, which would (of course) lead them to good ol' Frodo Baggins (aka Mr. Underhill).
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"I think we dream so we don't have to be apart so long. If we're in each others dreams, we can be together all the time." - Hobbes of Calvin and Hobbes
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