tolkiens legendarium - Why does Gollum calling the ring his own not alert Sauron?

This (and others eg: this ) answer states that Frodo claiming the ring alerted Sauron immediately of its position, which makes sense: How else would Sauron realize where the Ring and Frodo were at that exact moment. And it also explains why Sauron did not know where the ring was beforehand.

However there is a problem with this: Why did Sauron not find out about where the ring was when it was in Gollum's possession? Gollum obviously claimed the ring as his own (he even dreamt about becoming Gollum the Great, unfortunately I'm not able to find the passage in the books). In addition Bilbo claims it as well (right before he leaves it for Frodo), without alerting Sauron (Sauron learns about the Shire from Gollum, not through some form of comunication with the ring)

Is there an in canon explanation, or have I simply misunderstood something?

Answer

Joseph R's answer is a good one -- by wearing the Ring, Frodo did make himself more noticeable to the Ringwraiths nearby, but not to others far away. So proximity matters. But the most important proximity is that of the Ring itself to Mount Doom in particular. The Ring becomes more powerful as it approaches Sammath Naur, the place of its forging and its unmaking.

The last volume shows this happening in multiple ways: the Ring tempted Frodo more strongly as they approached Mount Doom; it became more burdensome to Frodo; and it may even have changed size and become heavier.

Furthermore, I don't think the increase in power is linear. I believe the power of the Ring increases rapidly -- say, inversely proportionally to square of distance -- as it approaches Mount Doom. If that's correct, then the power of the ring would have been many orders of magnitude higher when Frodo put it on at the end of the journey than at any earlier time. Even when Sam put it on at Cirith Ungol, they were still many miles away, and the ring would have been far less powerful.

I don't have a copy of the book handy to confirm this, but I believe this interpretation explains almost every detail in the book concerning Sauron's perception of those who wear the ring. It doesn't explain why Sauron perceives Frodo on Amon Hen, but the properties of the Seat of Seeing provide an alternative explanation in that case.

At no point in many hundreds of years had the Ring been so close to the place of its forging. Frodo's actions on Mount Doom were entirely unprecedented -- nothing that Gollum did with the Ring when he possessed it could have had the same effect so far from Sammath Naur. While the fact that Frodo "claimed" the Ring may have made his actions still more noticeable to Sauron, I think the Ring's proximity to Mount Doom was the primary factor.