Sam and the Ring

[Raynor]

Sauron really didn't have much "mental" power to "feel out" the Ring and its presence, even when it is right on his borders.

I believe he does have it, he was able to feel Frodo from very far away, at Amon Hen; he would have pinpointed him, if Gandalf hadn't stepped in (as he later recounts):

Originally Posted by The Breaking of the Fellowship, FotR

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.