The Tavern

Does the ring of stealth actually do anything? Yes, obviously it does something, increasing one's stealth; but by how much? Just empirically, it doesn't appear to make much difference in my rate of successful sneakiness. Also, is the effect stackable with multiple rings?

It's stackable. Sneaking is so random that being able to "notice" additional stealthiness would be near-impossible for a human. It adds a fixed amount to your stealth, so the more stealth you have, the less useful it is (as a percentage of how effective your stealth is already) "How much" is exactly one full + on the stealth bar (50 stealth points) without anything to compare it to, that's not very useful, but if you're paying attention to how much stealth skill it takes to get one more plus, you at least have something to compare it to.

According to knowledge bots, each level of stealth skill gives you 15 stealth points, while each ring of stealth gives you 50 points. So the ring gives as many stealth points as 3.3 levels of stealth skill.

It's a strong ring early game. I'd replace it past the opening game, unless your entire build is focused around stealth and you don't have any other better options.

Svendre is right, it's mostly something you'll find useful earlier in the game. When you get a decent amount of stealth, you can just dump the ring and carry more useful stuff instead, because its effects will be limited when compared to your skills at this point.