It does, in fact, turn humans invisible too. Isildur being the obvious example, but even the nine rings given to humans had that effect, shifting them to the spiritual / unseen world. That's a whole different ramble, but for now, let's sum it up that there is an unseen world not everyone can sense and influence, but the Maiar (including Sauron) are inherently spiritual beings that took physical shape in the seen world in order to interact with it.
For Sauron, so much of his power was poured into the One Ring that the continued existence of the ring meant that he could survive destruction of his physical form and eventually take shape again¹, though its loss in the War of the Last Alliance obviously robbed him of much of his strength and he had to hide for a long time to slowly regain his strength and renew his efforts.²
Through the Ring, Sauron had also dominated the nine human Ring bearers and bound them to him, moving them into the spiritual world. As his form was destroyed, so did they lose theirs. As he returned, so did they.
The reason they could still "see" Frodo is that they were attuned to the unseen and could sense him there, with their power over it manifesting in them stabbing his physical form even though it was invisible to mortal eyes.
There is still the question of the Dwarven rings. They were forged first, and it's possible they weren't as refined yet, though the dwarves are also described as more resilient at resisting the dominating effect. My guess is that the fact they were created by Aulë, Smith of the Valar, rendered them less susceptible to the craft of a lesser spirit (Sauron), but I have no evidence.
1: This paragraph and the following one originally read that Sauron could no longer take shape without the ring. I stand corrected on that, see the responses.
2: After his first destruction during the Fall of Numenor, his spirit managed to escape with the ring. 110 years later, he had enough strength to launch a strike against the nascent Gondor and start another war that lasted 13 years. He was destroyed again, this time losing his ring, and it took him a thousand years to become active again.Per the correction, he will probably have regained his form, though he was still too weak to fully reveal himself and start another open war for two thousand more years.
One of my colleagues will even occasionally ask me "Heading for a smoke, wanna come along?" I just love chatting with him, I'll try to stand upwind so I don't catch as much second-hand smoke, he gets some company too, everyone's happy.