I never really understood this in the books. How could Sauron not know that their intent was to destroy the ring? It is made clear that Sauron can see Frodo's location when he puts on the ring. Frodo puts on the ring at various points throughout the series, each time increasingly close to Mordor. How could Sauron have not considered this to be their intent?
We understand early on that employing the Ring is too hazardous to contemplate, and will simply lead to a different kind of doom, at best.
Sauron is worried that Saruman or Gandalf or Galadriel or Elrond or Aragorn could successfully command the full powers of the Ring, gather together the might of Men, and beat him down to nothingness, so that he must suffer another 3000 years of powerlessness before trying again.
Sauron may suspect that another could not do as much. But how could he know with certainty? He only has fifth hand reports of Isildur's tenure as the bearer of the Ring. The Ring was intended as a powerful tool for Sauron himself -- its nature in the hands of others may be as much a mystery to Sauron as anyone else.
The drawbacks of the Ring that we become so intimately familiar with over the course of three books are all perks in Sauron's mind.
The approach of the Ring is thus the approach of an adversary, using Sauron's own power against him. Sauron was probably wondering why Aragorn didn't brandish the Ring the moment he arrived at the Black Gate to challenge him.
