As far as we know, Sauron cannot detect the presence of the Ring, or even someone putting the Ring on. He only becomes aware of someone if they put it on and claim it as their own possession, while they are relatively close to him. This is why - in the books - Sauron only becomes aware of Frodo when he puts the Ring on in Mount Doom and says "The Ring is MINE!".
The books don't tell us much about this, but it seems that Sauron realizing that Frodo was inside Mount Doom with the Ring is based on a combination of factors:
- He claimed the Ring as his own personal property.
- He then wore the Ring.
- He was very close to Sauron when he did so - virtually on Sauron's doorstep.
Gollum claimed the Ring, and frequently wore it, for centuries, long before Frodo became a Ringbearer. But he was very far away from Sauron, deep beneath a mountain, for most of that time, and Sauron was very weak at the time as well. Gollum almost certainly didn't know where the Ring came from, or what it was, or who it belonged to, so he was simply claiming possession of a neat Ring that made him invisible and junk.
Bilbo also didn't know what the Ring was, who it really belonged to, or where it came from. He, too, was just taking possession of a neat invisibility ring. And again, when he first got it, Sauron was fairly weak, and the Ring was in the Shire, far away from Sauron, for most of the time that Bilbo owned it.
Frodo, on the other hand, knew what it was, who it belonged to, and where it came from. Unlike Gollum and Bilbo, he knew what he was doing when he claimed possession of the Ring. And unlike Gollum and Bilbo, Frodo claimed it and wore it right in Sauron's face.
The movies are very different from the books. In the movies, Sauron seems to perk up and take note whenever Frodo wears the Ring. This is not the case in the books.
In the books, Sauron only becomes aware of Frodo when he claims the Ring inside Mount Doom and then puts it on.
Sauron putting his own power into the Ring doesn't change anything. To use an analogy, your blood gives you the power to live, more or less. Let's say you donated blood to the Red Cross, and they used it in a blood transfusion. Do you really think you'd be able to identify the recipient if you passed them on the street? Of course not. The fact that some of your life force is coursing through their veins doesn't mean you would know who they are or where to find them. You would only know who they were if you saw the transfusion taking place.
When Frodo claimed the Ring and put it on his finger while in close proximity to Sauron, Sauron was watching the transfusion take place, metaphorically speaking.