The essay "Notes on Motives" sheds some light on this; Sauron was a "taking off the training wheels moment" for Men:
It is very reasonable to suppose that Manwë knew that before long (as he saw 'time') the Dominion of Men must begin, and the making of history would then be committed to them: for their struggle with Evil special arrangements had been made! Manwë knew of Sauron, of course. He had commanded Sauron to come before him for judgement, but had left room for repentance and ultimate rehabilitation. Sauron had refused and had fled into hiding. Sauron, however, was a problem that Men had to deal with finally: the first of the many concentrations of Evil into definite power-points that they would have to combat
History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (iii)
There may also have been a certain amount of foreknowledge involved; Tolkien earlier writes of Manwë:
Manwë was the spirit of greatest wisdom and prudence in Arda. He is represented as having had the greatest knowledge of the Music, as a whole, possessed by any one finite mind; and he alone of all persons or minds in that time is represented as having the power of direct recourse to and communication with Eru. He must have grasped with great clarity what even we may perceive dimly: that it was the essential mode of the process of 'history' in Arda that evil should constantly arise, and that out of it new good should constantly come.
History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (iii)
Although this was written of Manwë's response to Morgoth, it presumably applies equally well to Sauron. Manwë has certain "inside information."
Having said all that, it is worth noting how similarly the struggle against Sauron parallels the struggle against Morgoth:
- Super-powerful Nemesis with a massive hate-on for the entire race
- The race keeps the Nemesis more-or-less in check through military force
- The war is ultimately a losing battle; they can't defeat the Nemesis on their own merits, they can only stall him
- The Nemesis has externalized much of their power, making them vulnerable to defeat (Sauron into the Ring, Morgoth into the Matter of Arda)
- The Nemesis is finally defeated with the assistance of an external power (the Host of Valinor and the War of Wrath defeat Morgoth, Ilúvatar defeated Sauron)
The response of the Valar in the Third Age is remarkably similar to their response in the First. Sauron is the Nemesis of Men, in the same way Morgoth was the Nemesis of the Elves, and each race must ultimately struggle for themselves (even if, as Tolkien is fond of pointing out, Evil is not finally resistable by Good).