From what I've read, the Elves hid their rings while Sauron still held the One Ring. The 7 rings for the Dwarf Lords did not respond as Sauron had intended them to.
In the Second Age Sauron gave the Seven to various Dwarf-lords (though the Dwarves of Moria maintained a tradition that the ring given to Durin III came directly from the Elven smiths). Gandalf mentions a rumour that the seven hoards of the dwarves began each with a single golden ring so perhaps they do have some sort of ability to protect the bearer (otherwise he might not have been so lucky in his quest for wealth). The main power of the Seven on their wearers was to excite their sense of greed, and the Dwarves used their rings to increase their treasure. The wearers did not become invisible, did not get extended life-spans, nor succumb directly to Sauron's control – though he could still influence them to anger and greed.
But Sauron recovered three rings from the Dwarves, the last from Thráin II during his final captivity in Dol Guldur some years before the beginning of The Hobbit. The remaining four, according to Gandalf, were destroyed by dragons (consumed according to some sources). Perhaps one or two were discovered years later, after all the dragons will have decayed or been gutted for food etc.
Until the Council of Elrond, the Dwarves did not know that Thráin had held the ring of Durin's line and had lost it to Sauron. They thought instead that it might have been lost when Thrór was killed by Azog in Moria. One of the motivations for Balin's doomed expedition to Moria was the possibility of recovering the ring. Also Sauron's messenger attempted to bribe the Dwarves of Erebor for news of Bilbo (the last known bearer of the One) with the promise of the return of the remaining three of the Seven. Plenty of fun to be had with scenario or character special rules
