It’s a bit more complicated than invisibility...
This occurred to me and I feel it’s worth posting since I’ve never seen any talk on this?
The One Ring doesn’t make you invisible.
Why would it? Seriously, what purpose on Eru’s green earth does that serve? Sauron forging his ring of power in the heart of a volcano, thinking to himself, ah yes, invisibility would be a good trick to build into this thing! No. Cause you know what? Sauron’s ring does not make him invisible. And he certainly did not intend for anyone else to ever have it. So what’s it really doing?
Two words: Dimensional shift.
I believe that when mortals put on the ring, they experience a dimensional shift in which they are pulled (stretched, transported) into a higher dimension, the plane on which the true spirit forms of the Ainur (and wraiths) exist. This would effectively render them invisible to those on lower dimensions, but the wearer would be able to view them with altered enhanced perception. Such as the effects we witness as described by those who have worn the ring. Especially well portrayed in the films is the ability to see the souls of others, particularly the ringwraiths (the battle on Weathertop is a good example, as well as even in Battle of the Five Armies when Bilbo is in Dale), black and white shadowy souls clear as day but invisible to the naked eye, as they exist on a different dimensional plane. It’s quite possible to me that the ëalar of the Ainur are in a higher dimension than that of mortal fëar, but that’s beside the point. They’re at least a couple dimensions removed from our reality, and thus invisible until one puts on the ring.
Now, why would the ring have this power? I think, if I recall correctly, that Tolkien at one point did state that it was not intentional, that it was a byproduct of its making. Again, it does not turn Sauron invisible - it wouldn’t, he already exists on that plane. Mortals are bound to their bodies and so would not be able to perceive that higher dimension, but Ainur are not. I think most plausibly, this effect exists because Sauron infused a piece of his own soul into the one ring. The consequences of this are not well understood (it’s not like it’s a common practice) and we know in other ways, it is so strong in its desire to hearken back to its master, it can even influence the wills - a product of the souls - of those around it. I would not be surprised in the slightest if having a piece of Ainur ëala in an all-powerful object would result in the ability to bend reality to attempt to match the wearer to the properties of the owner. It would bring the wearer closer to Sauron, and allow him to perceive them, thus furthering its purpose to return to the whole from which it is a part.
Just a theory, obviously, but I find it odd that I’ve never seen anyone question the rather absurd notion of ‘magic evil ring makes you invisible!’ Tolkien set up so many intriguing questions, concepts, and possibilities with underlying scientific principles - or at least, consistent rules - that I am sure this fits into his framework.