Why were the keepers of the three Elf-rings not aware that the One Ring had been used?

Question

All twenty Rings of Power are linked, as we know. Further, we know from the text of the Lord of the Rings that the One Ring can confer the power to "know the thoughts" of the keepers of the other Rings, including the three Elf-rings. That Bilbo and Frodo did not have this ability is explained during Frodo's encounter with Galadriel (The Lord of the Rings, Book 2, Chapter 7: The Mirror of Galadriel):

'I would ask one thing before we go,' said Frodo, 'a thing which I often meant to ask Gandalf in Rivendell. I am permitted to wear the One Ring: why cannot I see all the others and know the thoughts of those that wear [the Elf-rings]?'

'You have not tried,' she said. 'Only thrice have you set the Ring upon your finger since you knew what you possessed. Do not try! It would destroy you. Did not Gandalf tell you that the rings give power according to the measure of each possessor? Before you could use that power you would need to become far stronger, and to train your will to the domination of others.'

Although here Frodo and Galadriel are explicitly talking about Vilya, Narya, and (in particular) Nenya, it's reasonable to assume that this "perception" would extend to the sixteen other Rings of Power.

However, this "perception" does not appear to be a case of simple one-way mind-reading (The Silmarillion, Book 5: Of the Rings of Power and the Third Age):

Now the Elves made many rings; but secretly Sauron made One Ring to rule all the others, [...] And while he wore the One Ring he could perceive all the things that were done by means of the lesser rings, and he could see and govern the very thoughts of those that wore them.

But the Elves were not so lightly to be caught. As soon as Sauron set the One Ring upon his finger they were aware of him; and they knew him, and perceived that he would be master of them, and of an that they wrought. Then in anger and fear they took off their rings.

Clearly, to some extent the keepers of the three Elven rings knew from the moment Sauron used it that the One Ring existed, and that Sauron was a threat to them, particularly so while Sauron wore the One and they wore the Three. (See note 1.)

So, why were the keepers of the Three unaware that the One had been found and, more importantly, used prior to Gandalf's discovery of the nature of Bilbo's magic ring? After all, it was used many, many times: Sméagol (Gollum) used the Ring extensively in various mischiefs and murders, both before and after his time under the Misty Mountains; Bilbo used it several times in close proximity (ranging from several miles to just a few yards) of Gandalf who bore Narya; Frodo used it five times during his quest; and Sam even wore the ring briefly. Meanwhile, Gandalf, Elrond, and Galadriel likely never removed their Rings through the entire timeline of published texts of The Hobbit and The Lord of the Rings. Would not the use of the One have been noticed by these three?

Related question(s) and side-notes:

• Ring-bearers — who can notice what about whom and when?: The only extensive answer to this question seem to support the notion that two Ring-keepers can sense various things about each other so long as one of the two Rings is the One Ring.
• Note 1: Although no such awareness was mentioned about the Men and Dwarves who wore the Nine and Seven (despite the dooms that befell them, in the end), this can be likely be attributed some combination of: Sauron's deceit (they were given and accepted as gifts, after all); the Sixteen being different from the Three or the One (being forged by Celebrimbor with Sauron's guidance, unlike the Three, forged by Celebrimbor and the Elves without Sauron's hand, or the One, forged by Sauron alone); or the wisdom of those who bore the Rings (all of keepers of the Elven rings at or near the time — Gil-galad, Galadriel, and Celebrimbor — were Deep Elves: Elves born in Aman during the Years of the Trees. Of course, others such as Gandalf (a Maiar) and Elrond (the half-elven) kept rings later.).

Answer 1

I don't think any of the hobbits (Smeagol/Gollum, Bilbo, Frodo, Sam) ever "used" the ring in the sense meant by the text you quote from Silmarillion [*]. The one ring was created to perceive and control the thoughts of others, as it states. When Sauron initially used it in an attempt to do exactly that to the wielders of the Three, they were immediately aware of it. Turning the ring-wearer invisible was a (unintended?) side-effect that apparently did not depend on "using" the ring, only on "wearing" it.

This is pretty much what Galadriel says in the text you quote - "you have not tried", meaning "you have only put the ring on your finger, not tried to use it".

[*] Except, of course, at the very end standing by Sammath Naur when Frodo puts the ring on his finger and "claims it as his own". At that moment Sauron did indeed sense Frodo's use of the ring immediately. One could speculate that the keepers of the Three sensed it also, but I don't know of any textual evidence.

Answer 2

The One Ring was the master of the others. It was not, in any way, subject to their lesser powers. This means, logically, that the One was made so that it could be used without its use automatically being detected by the other Rings. Whoever bore the One Ring could use its lesser powers without alerting any of the connected Rings to its activity. When Sauron was making (and presumably testing) his Master Ring, the other Ring bearers were apparently totally unaware if his machinations, until he reached out to take control of the other Nineteen Rings. Only when the One Ring's power was actively being used to interfere with the functioning of the other Rings did the wielders feel its presence.

The answer to why the remaining free wielders of the other Rings did not detect its use is fairly simple. (There were at least four Rings operating outside Sauron's control at the time when Deagol found the One by the Gladen Fields: the Rings of Air, Fire, and Water; and the Ring of Durin. The Longbeards claimed to have received their ring directly from the elves of Hollin, which is plausible since the House of Durin was based in Moria and closely allied to Celebrimbor's people at the time. But however the House of Durin got their Ring, they stayed free of the Dark Lord's domination, and Sauron did not recover that last of the dwarven Rings until much later.) The powers being used by Gollum and later Bilbo were very limited. They used it to make themselves invisible and to heighten their own perceptions. These lesser powers had nothing to do with the other Rings, and so they could be used undetectably.

There is one obvious objection, which is that when Frodo donned the Ring at Weathertop, he became more visible to the Ringwraiths attacking the camp. However, I think this is actually more about the mechanism by which the Ring conferred invisibility than direct communication between the Rings. The Ring made its wearer invisible to normal sight by shunting the wearer into the spirit realm. This made Frodo unseen to normal eyes, but the Nazgul were already in the spirit realm themselves, having worn their own Rings so long that they had were permanently faded from the normal world. So it is not coincidence that the wielders of the lesser Rings could detect Frodo when he used the One, but it is not a matter of communication between the Ring wielders either.