I agree with rutslegolas about Isildur and Sauron not being able to do anything about it. I think the part with Gollum and Bilbo is that when they said "It is mine," they really didn't understand completely what it was they were saying. When Frodo said it he knew full well about Sauron and the Ring's powers. He was in fact claiming it, challenging Sauron to an extent. Also when Gollum and Bilbo had the Ring Sauron was still growing again and he had thought the Ring was lost forever. He didn't start looking for it until Frodo had gotten the Ring when he realized the Ring wasn't lost.
I'm not sure about my timing on that. Sauron may have found out about the Ring still being there when Bilbo had it. Someone feel free to correct me if I'm wrong.
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