Despite Gandalf's words here, there is no evidence that the Rings of Power (other than the One) exhibited this behaviour.
In fact, taken as a whole, Gandalf's statement is already provably false; we have several examples of Rings being voluntarily given up by their owners:
- Celebrimbor initially dispersed the Three to Galadriel and Gil-Galad1
- Gil-Galad gave away the two he was given (Vilya to Elrond and Narya to Círdan)
- Círdan gave Narya to Gandalf
- Thrór gave his Ring to his son Thráin
Even if we assume that Gandalf is only partially wrong here, we have no basis for speculation one way or the other; there's simply no writing discussing this idea further.
Personally, I suspect that Gandalf's line is a holdover from earlier versions of the Legendarium, where the distinction between the One and the other Rings of Power was not as clearly defined as in later drafts; for example:
[W]hat has all this to do with me and the Ring?'
'It is the only Ring left,' said Gandalf.
'In the ancient days the dark master made many Rings, and he dealt them out lavishly, so that they might be spread abroad to ensnare folk. [...] Men had three rings, and others they found in secret places cast away by the elf-wraiths: the men-wraiths are servants of the Lord, and they brought all their rings back to him; till at last he had gathered all into his hands again that had not been destroyed by fire - all save one.
History of Middle-earth VII The Return of the Shadow Part 1: "The First Phase" Chapter III: "Of Gollum and the Ring"
1 In fairness, it's said in Unfinished Tales that Celebrimbor was counselled by Galadriel to send the Three away from Eregion:
Galadriel counselled him that the Three Rings of the Elves should be hidden, never used, and dispersed, far from Eregion where Sauron believed them to be. It was at that time that she received Nenya, the White Ring, from Celebrimbor, and by its power the realm of Lórinand was strengthened and made beautiful; but its power upon her was great also and unforseen, for it increased her latent desire for the Sea and for return into the West, so that her joy in Middle-earth was diminished. Celebrimbor followed her counsel that the Ring of Air and the Ring of Fire should be sent out of Eregion; and he entrusted them to Gil-galad in Lindon.
Unfinished Tales Part 2: "The Second Age" Chapter IV: "The History of Galadriel and Celeborn"
It may be that she had to "persuade" him in a similar fashion to how Bilbo was "persuaded" by Gandalf to leave the One with Frodo.