My guess is that your second guess is probably closer to the truth out of the two. The One Ring was the master Ring; it controlled the others and could bring them together to the person who held it- in this case, Sauron.
Even though Celebrimbor forged the Three Rings without any direct aid from Sauron (apart from the lore and crafts Sauron taught him), the One Ring still controlled the Elven Rings and all the other Great Rings (the Nine and the Seven). Sauron placed a lot of his own power into the One Ring in order to be dominate the other Rings. Therefore, when he is power is dissipated, so too would the power of all the other Rings, including the three. Let me give you a basic example- if something happens to your local electricity generator, then there's no power to light up the lightbulbs.
Thus, the Three Rings - along with the rest of the Great Rings - really became tied to the power of the One and once that source was gone, well, so was their power also. This leads to the question of why then could the Elves use the Rings without being detected while the One was still in existence? Well, two reasons-
a) Sauron didn't possess the One Ring at that time, so he couldn't 'bind' all the Rings to his power. Remember, as soon as Sauron spoke the words inscripted on the Ring in the Sammath Naur that Celebrimbor was aware of his plans and took off the Rings immediately.
b) Frodo, Gollum, Isildur and Bilbo didn't try to find the Three Rings and didn't have the power to, albeit that the Ring gave Frodo the 'power' to see Galadriel's Ring in Lothlorien.
Anyway, I'm diverging, so I'll leave you with a quote that doesn't really explain much, but is still interesting, coming from one of the Wise-
Quote:
'But what then would happen, if the Ruling Ring were destroyed, as you counsel?' asked Gloin.
'We know not for certain certain,' answered Elrond sadly. 'Some hope that the Three Rings, which Sauron has never touched, would then become free, and their rulers may heal the hurts of the world that he has wrought. But maybe when the One has gone, the Three will fail, and many fair things will fade and be forgotten. That is my belief.'
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In the end, that's what happened.