The Nazgul did not wear their rings. Sauron held their rings. Their invisibility was a permanent state of their flesh, not their belongings.
In a letter from circa 1963 Tolkien says explicitly that Sauron held the rings:
They would have obeyed . . . any minor command of his that did not interfere with their errand — laid upon them by Sauron, who still through their nine rings (which he held) had primary control of their wills . . . — The Letters of J.R.R. Tolkien, Letter 246
They were by far the most powerful of his servants, and the most suitable for such a mission, since they were entirely enslaved to their Nine Rings, which he now himself held . . . — Unfinished Tales, p. 338
As I mentioned in a different post, the Nine, as living Men, were vain and arrogant who, while they were living, wore the rings because of the power they gave them as kings, sorcerers and warriors. Sauron would want them to wear the rings all the time to ensure their transition into his servants. I would think their vanity would have the rings being visible all the time, only further increasing their fame and notoriety. They were not in hiding, nor were they afraid of their enemies.
Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had, as it seemed, unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men; but too often they beheld only the phantoms and delusions of Sauron. And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore and of the domination of the One which was Sauron's. And they became forever invisible save to him that wore the Ruling Ring, and they entered into the realm of shadows. The Nazgûl were they, the Ringwraiths, the Úlairi, the Enemy's most terrible servants; darkness went with them, and they cried with the voices of death. — The Silmarillion, "Of the Rings of Power and the Third Age", 346
Emphasis is mine, but it answers the question, they possessed the power of choice as to whether they would be invisible or not. And I believe the One Ring, in the right hands would also give the power of choice as to whether the wearer could become invisible. The wearer probably needed the power or the skill to control the ring, of which only Sauron seemed to have both.
