Gandalf says in "The Shadow of the Past," "Well, as for the name, Bilbo very foolishly told Gollum himself; and after that it would not be difficult to discover his country, once Gollum came out."
Furthermore, Gandalf says that Gollum followed Bilbo's trail through Mirkwood and even into Esgaroth. Also, Gollum knew that Bilbo was a hobbit, so all he really had to do is overhear someone say something like "the Shire's a place where a lot of Hobbits live" and he would know the name of the place.
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"He was sent by a mere prudent plan of the angelic Valar or governors; but Authority had taken up this plan and enlarged it, at the moment of its failure."
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