I have always thought that the ring was a kind of amplificator of personal power (magical and mundane), which is explained by Galadriel when she refuses Frodo's offer (everybody would love me blabla..) and by dreams like Sam's cited previously (albeit the delusive effect of the ring).
This theory goes well with the powers of a Galadriel (first born of the Noldor) or a Gandalf (Istari or Maia in human form). They would have developed tremendous powers with the ring (exactly like Sauron). But both are also aware of its corrupting nature and pass the test (they refuse the gift).
If the power of hobbits is clearly "discretion" (as Gollum) and simplify the idea of invisibility associated with the ring: What about a great warrior like Isildur (invisible not super warrior ???). My explanation is that the rings over their amplifying powers put you a little bit more in the "spirit" world (The source of power, the original song of the Ainur).
The Humans & Hobbits have souls, not spirits like the aforementioned: so they "shift": if you are mortal you gain presence in the spirit world and leave nearly no trace in the physical world (The invisibility is not a "perfect one" as explained in The Hobbit).
But beings of power, having spiritual & physical presence, are in both worlds so they don't become invisible.
This also explains the "amplifying" power of the other rings (the 3 or the 7): riches for the dwarves (but as the rings are tainted this leads to greed), security and protection of their realms & kinds for the elves.
The magic of Middle Earth is very subtle (not of the D&D Fireball style), but Gandalf wearing the ring of Fire is still the best Pyromancer(Anor) of his time ;)
In my opinion all the pieces of the puzzle fit !
By the way Sauron & Gandalf are both Maia "issued" from Aulë. One corrupted by Morgoth the other not. Both are using their communication skills to achieve their goals, one with empathy the other with manipulative evil.
No need to wonder what Gandalf's intelligence, empathy and secrecy would have given with the corruption of the One...
One Ring to rule them all !
Yes, all the rings touched by Sauron were linked to the One (except the 3), which simplified the control he sought over the main rulers of the time (who received those rings).
The rings were also built for a specific racial trait (or race) as the poem goes...
Sauron's effort to get the last of the 7 for him when is plot was thwarted by Dwarf resilience to magic is also a good indicator of their real "wealth generating effect" to rebuild his new Mordor.
Hope you liked my prose ;)