We don't get much of anything in the Lord of the Rings about Sauron's actual physical form, but there's nothing to my mind that suggests this is anything like the keen sight of Legolas.
Remember that Sauron, being a Maia, probably had powers (as for example Melian did in the First Age) which extended beyond the purely physical. Melian's abilities allowed her to ward strangers off from a distance (creating the Girdle of Melian around the forest of Doriath); Sauron seems to have been able, perhaps not to literally see but to understand and detect events at a distance.
There are a few times we see Sauron's sight; the first is at the end of The Fellowship Of The Ring:
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.
Here this seems to be much more a presence or a "will" that Frodo feels. Sauron doesn't appear to know where he is exactly, but can focus his will on the general area where he feels Frodo's (magically-enhanced) gaze coming from. The encounter with Sauron is described in terms of touch as much as in terms of sight.
The other times I can remember any sort of encounter with Sauron are in the Return of the King.
One moment only it stared out, but as from some great window immeasurably high there stabbed northward a flame of red, the flicker of a piercing Eye; and then the shadows were furled again and the terrible vision was removed. The Eye was not turned to them: it was gazing north to where the Captains of the West stood at bay, and thither all its malice was now bent, as the Power moved to strike its deadly blow; but Frodo at that dreadful glimpse fell as one stricken mortally. His hand sought the chain about his neck.
This is the closest we ever get to seeing something like what Peter Jackson had in the movies. Note, though, that Eye is capitalized. We're not talking, it seems, about Sauron's literal eye; perhaps about some manifestation of his searching presence - a great evil, which nearly strikes down Frodo.
We see this sort of manifestation once more when Frodo claims the ring:
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.
Again, this seems to be not physical sight but some sort of physical or somewhat physical manifestation of a strong will searching for something.
Sauron certainly did have, and use, a palantir at Barad-dur; but we see little evidence of him actually using it; his encounters with Pippin or Aragorn are the only time I can see that. He may or may not have directly encountered Denethor as Denethor was using the palantir of Minas Tirith; certainly he realized what Denethor was doing. But I don't think that most of his knowledge of events far away was gained that way. The palantir could give views of places in general, without needing to communicate with other stones, but this would have been inconvenient, as Sauron would have had to walk around the stone to look in different directions.