The wearers of the Three Elvish Rings of Power could detect when Sauron gained or lost the ring, as they did when he forged The One Ring and placed it upon himself:
As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and preceived that he would be master of them, and of all they wrought. Then in anger and fear they took off their rings. - ("Of the Rings of Power and the Third Age," Silmarillion)
Whereupon they removed their rings until they found out he been struck down and the ring removed from him.
We could assume that the wearing of the rings was to detect if Sauron gained the ring back and thereby plan for the future. Middle-Earth does not have a wonderful postal or newspaper system so this could give them information they needed months in advance of when they'd otherwise gain it.
From the quote we can assume that it wouldn't be an immediate transformation into slavish domination by Sauron so the removal of the rings would allow for them to continue as they were, if but without their power.
If they did continue to wear their rings, they would've likely have ended up in a form of slavery similar to that of the Ringwraiths, able to subvert commands if they truly desired to but not able to directly resist. In a similar way to how the Ringwraiths would've acted if Frodo had managed to keep possession of the Ring:
I think they would have shown 'servility'. They would have greeted Frodo as 'Lord' ... Until Sauron himself came ... Sauron would not have feared the Ring! It was his own and under his will. (Letter 246)