The ring doesn’t as much make you invisible as it transports you into the shadow realm. Sauron, however existed in both the real world and the shadow realm, so there was no transporting needing to happen there.
The ring’s primary power is control over the other rings, and it seems like the shadow realm is part of how that works (“…and in the darkness bind them”). I’m guessing Frodo is overwhelmed by the power of the ring and is taken to the shadow realm, whereas Sauron has more control and can stay in the real world.
Other races may have different effects. The elves have some amount of magical ability, so maybe they can stay in the real world too. Maybe the dwarves would concoct some device to block part of the ring’s power to stay in the real world. We don’t know, the only examples we have are halflings and Sauron.
The ring doesn’t as much make you invisible as it transports you into the shadow realm. Sauron, however existed in both the real world and the shadow realm, so there was no transporting needing to happen there.
The ring’s primary power is control over the other rings, and it seems like the shadow realm is part of how that works (“…and in the darkness bind them”). I’m guessing Frodo is overwhelmed by the power of the ring and is taken to the shadow realm, whereas Sauron has more control and can stay in the real world.
Other races may have different effects. The elves have some amount of magical ability, so maybe they can stay in the real world too. Maybe the dwarves would concoct some device to block part of the ring’s power to stay in the real world. We don’t know, the only examples we have are halflings and Sauron.