The Ring and Its GPS Tracker

[Puddleglum]

I think part of the answer is given in RoTK, "The Tower of Cirith Ungol" as Sam crosses the summit of the pass (removing the Ring from his finger as he does so)

His [Sam's] thought turned to the Ring... No sooner had he come in sight of Mount Doom than he was aware of a change in his burden.
As he drew near the great furnaces where, in the deeps of time, it had been shaped and forged, the Ring's power grew...
'He'd spot me, pretty quick, if I put on the Ring now, in Mordor'.

As far as Amon Hen, that was the Numenorean's "Seat of Seeing". The seat, coupled with the Ring, had already combined to let Frodo's gaze carry even to Barad Dur.

While we don't know the specifics of "how" that worked, the fact that it "DID" work, opens the possibility that the combination also allowed Sauron to "look" back. In the event, that took time and Frodo managed to remove the Ring before Sauron's back-trace could be completed.