There's no reason to believe Sauron could detect the Ring or when anyone was using the Ring, except under exceptional circumstances:
One time almost, on the Hill of Seeing, which was a special place - considering Frodo was able to see an incredible amount of places and happenings:
Then at last his gaze was held: wall upon wall, battlement upon battlement, black, immeasurably strong, mountain of iron, gate of steel, tower of adamant, he saw it: Barad-dûr, Fortress of Sauron. All hope left him.
And suddenly he felt the Eye. There was an eye in the Dark Tower that did not sleep. He knew that it had become aware of his gaze. A fierce eager will was there. It leaped towards him; almost like a finger he felt it, searching for him. Very soon it would nail him down, know just exactly where he was. Amon Lhaw it touched. It glanced upon Tol Brandir - he threw himself from the seat, crouching, covering his head with his grey hood.
Barad-dur was quite some distance away*, not to mention behind a mountain range. Frodo had some degree of supernatural sight on the Hill of Seeing. It makes 'magical' sense that to see everything, Frodo must to some degree make himself visible, particularly to Palantir-wielders like Sauron.
The only other instance was this:
And far away, as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-dûr was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him, and his Eye piercing all shadows looked across the plain to the door that he had made; and the magnitude of his own folly was revealed to him in a blinding flash, and all the devices of his enemies were at last laid bare. Then his wrath blazed in consuming flame, but his fear rose like a vast black smoke to choke him. For he knew his deadly peril and the thread upon which his doom now hung.
Frodo not only put the Ring on, he claimed it as his own with full knowledge of the Ring's power and what it was, and of the consequences, in 'the very heart of his realm', where the Ring was itself forged. This is clearly a magically significant event in a magically significant place.
There's no reason to believe that elsewhere and under normal circumstances that Sauron could detect people wearing the Ring. Indeed, if this was possible, Gandalf and Aragorn's feint would have been a tremendous failure, because it depended on Sauron believing that Aragorn had and was wielding the One Ring.
‘As Aragorn has begun, so we must go on. We must push Sauron to his last throw. We must call out his hidden strength, so that he shall empty his land. We must march out to meet him at once. We must make ourselves the bait, though his jaws should close on us. He will take that bait, in hope and in greed, for he will think that in such rashness he sees the pride of the new Ringlord: and he will say: “So! he pushes out his neck too soon and too far. Let him come on, and behold I will have him in a trap from which he cannot escape. There I will crush him, and what he has taken in his insolence shall be mine again for ever.”
Sam is certainly no expert, but he believes he is only in danger wearing the Ring inside Mordor.
‘And anyway all these notions are only a trick,’ he said to himself. ‘He'd spot me and cow me, before I could so much as shout out. He'd spot me, pretty quick, if I put the Ring on now, in Mordor.
It is clear that Sauron was totally ignorant of Sam's Ring use on the very outskirts of Mordor:
Now the flicker of approaching torches and the clink of steel ahead was very near. In a minute they would reach the top and be on him. He had taken too long in making up his mind, and now it was no good. How could he escape, or save himself, or save the Ring? The Ring. He was not aware of any thought or decision. He simply found himself drawing out the chain and taking the Ring in his hand. The head of the ore-company appeared in the Cleft right before him. Then he put it on.
The world changed, and a single moment of time was filled with an hour of thought. At once he was aware that hearing was sharpened while sight was dimmed, but otherwise than in Shelob's lair. All things about him now were not dark but vague; while he himself was there in a grey hazy world, alone, like a small black solid rock, and the Ring, weighing down his left hand, was like an orb of hot gold. He did not feel invisible at all, but horribly and uniquely visible; and he knew that somewhere an Eye was searching for him.
The danger in using the Ring prior to their arrival in Mordor was not that Sauron would detect them (he would eventually detect an overt use, if say one person suddenly appeared leading all the others in a united front with absolutely subservient armies), but that trying to use the real power of the Ring would break Frodo.
'Only thrice have you set the Ring upon your finger since you knew what you possessed. Do no try! It would destroy you. Did not Gandalf tell you the rings give power according to the measure of their possessor? Before you could use that power you would need to become far stronger, and train your will to the domination of others.'
Note though, that at the end, Frodo was quite possibly using the power of the Ring to dominate the will of others - namely Gollum.
Sauron did put himself into the Ring - true. But in so doing, he externalized that power. It was bound up with him, but not a part of him like a limb that he knew where it was at at all times. That was the price Sauron paid for creating the Ring.
(all quotes from LotR)
(* It looks like Sam and Frodo traveled about another 500 miles from Amon Hen to Sammath Naur, but I'm not sure what the distance is as the crow flies.)