This question about the Ring's betrayal of Isildur brought up many different interpretations of the nature of the Ruling Ring and the extent of its ability to influence or even directly control events. Can the Ring make decisions on its own or is it merely a dangerous, but inanimate object?
You cannot press the One Ring too hard, for it is of course a mythical feature, even though the world of the tales is conceived in more or less historical terms.
The Letters of J. R. R. Tolkien, #211 To Rhona Bear, p. 279
Tolkien has written no texts that I could find about the exact nature or sentience of the Ring. The Ring seems to have several passive powers that would not require sentience, such as preventing decay and invisibility. Other powers are ambiguous, as they could be more of a reaction to its environment than a conscious decision, such as changing its size. Finally, there are other passages that would imply self-awareness and conscious decision making, but even those are unsatisfactory.
Let's define sentience first: "the capacity to feel, perceive, or experience subjectively". This is not to be confused with sapience: "the ability of an organism or entity to act with appropriate judgement". A simplistic example (that some might disagree with) would be this: a table is neither, an animal is sentient, a human is sapient.
What is the Ring?
And much of the strength and will of Sauron passed into that One Ring; for the power of the Elven-rings was very great, and that which should govern them must be a thing of surpassing potency; and Sauron forged it in the Mountain of Fire in the Land of Shadow.
The Silmarillion, Of the Rings of Power and the Third Age, p. 344
There is no clear explanation as to what transferring one's strength and will to an object means. The Ring is, as far as I know, the only occurrence of this kind of ability.
It may be that the Ring becomes a extension of Sauron, but that might require two-way communication at a distance, which clearly doesn't happen (or it would have been found easily).
The Ring could also have an independent will that mirrors Sauron's, but this is getting close to giving life to an inanimate object, which only Eru can do (as with the Dwarves, for example).
The only explanation I can come up with is that Sauron's "strength and will" have given the Ring passive powers that do not require sentience (preventing decay, invisibility, lust) and that any sentient-like behaviour is merely interpretation and anthropomorphization.
The Ring is often described as having a will of its own, but most of these quotes could also be hyperbole or sourced to an unreliable character or narrator. These are the most relevant quotes I could find.
"A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it.[...] It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him."
The Lord of the Rings: The Fellowship of the Ring, The Shadow of the Past, p. 73
Whether Gandalf is reliable here is not clear. He could also be anthropomorphizing it for the sake of discussion with a Hobbit. The Ring "decided things" by acting randomly, regardless of the owner's intentions.
And [Saruman] deemed that the Ring, which was Sauron's, would seek for its master as he became manifest once more; but if he were driven out again, then it would lie hid.
The Silmarillion, Of the Rings of Power and the Third Age, p. 362
This is Saruman talking at the White Council, where he is unreliable: "Curunír had turned to dark thoughts and was already a traitor in heart". This could also be interpreted as Sauron calling for the Ring rather than the Ring actively trying to get back to him.
So passed the first victim of the malice of the masterless Ring: Isildur, second King of all the Dúnedain, lord of Arnor and Gondor, and in that age of the World the last.
Unfinished Tales, The Disaster of the Gladden Fields, p. 357
This attributes malice to the Ring. I've included the quote because it is usually a sentient attribute. There's also another one about having "avenged its maker". Again, this sounds more like anthropomorphization and hyperbole to me than a statement about sentience.
Yet many have thought that the ferocity and determination of their assault on Isildur was in part due to the Ring. It was little more than two years since it had left his hand, and though it was swiftly cooling it was still heavy with his evil will, and seeking all means to return to its lord (as it did again when he recovered and was rehoused).
Unfinished Tales, The Disaster of the Gladden Fields, note 20, p. 366-367
Because of the weasel words "many have thought", I consider this to be an unreliable narrator. In any case, it doesn't imply consciousness: it might just be an Orc magnet at this point.
But as for throwing it away, that was obviously wrong. These Rings have a way of being found.
The Lord of the Rings: The Fellowship of the Ring, The Shadow of the Past, p. 79
I interpret this as a figure of speech and stretching the truth: the One Ring was found, but only after 2500 years.
The Ring is able to change its size. This happens so that it can fit on the wearer's finger, but it also seems to happen haphazardly:
"Though [Bilbo] had found out that the thing needed looking after; it did not seem always of the same size or weight; it shrank or expanded in an odd way, and might suddenly slip off a finger where it had been tight."
The Lord of the Rings: The Fellowship of the Ring, The Shadow of the Past, p. 62
Whether the Ring is changing its size consciously is unclear. Bilbo seems to imply that there was no discernible pattern as to when the Ring would do so, but that doesn't mean that he's right. It shrinked itself for Isildur, but then enlarged again two years later and got him killed.
There is an interesting quote from The Disaster of the Gladden Fields where Isildur intends to give the Ring to the "Keepers of the Three". At this time, this was Galadriel, Círdan and Elrond:
"I cannot use it. I dread the pain of touching it. And I have not yet found the strength to bend it to my will. It needs one greater than I now know myself to be. My pride has fallen. It should go to the Keepers of the Three." [...]
"My King," said Elendur, "Ciryon is dead and Aratan is dying. Your last counsellor must advise nay command you, as you commanded Ohtar. Go! Take your burden, and at all costs bring it to the Keepers: even at the cost of abandoning your men and me!"
Unfinished Tales, The Disaster of the Gladden Fields, p. 354-355
Minutes later, the Ring slips from his finger. If the Ring is sentient, then it might have heard Isildur and Elendur and decided that getting picked up by an Orc, or even being lost in the river would be a better fate. In this case, it could even be described as sapient, as it would be making a conscious decision based on new information.
But Isildur could just as well have been unlucky and experienced yet another random change in size, as described by Bilbo.
All citations from the Harper Collins editions
No, it was not. To grant the ring sentience and sapience would an act of creating original life. Aulë was not able to create dwarves who could move and act of their own accord without Aulë's bidding; Sauron, one of the lesser Ainur, could not create a ring that thought for itself either. True consciousness was the sole gift of Illuvatar.
So, we must conclude that what malice the ring had was merely an expression of Sauron's will. Sauron put part of himself into the ring, both his magical power and his evil hunger for might. The ring's dark "personality" was a reflection of the evil of its creator, who invested so much of himself in the artifact. Yet even though its personality was a piece taken from Sauron's, the dark lord could not actively communicate with the ring when it was away from him. His psyche was not split into two pieces that could remain in communication. We know this because, even at the greatest height of his return to power, at close of the Third Age, he could not even discern the location of the ring or who held it (until Frodo put it on in the heart of the Sammath Naur).
Gandalf certainly talks about the Ring as though it though it had a mind of its own. He tells Frodo:
‘A Ring of Power looks after itself, Frodo. It may slip off treacherously, but its keeper never abandons it. At most he plays with the idea of handing it on to someone else’s care – and that only at an early stage, when it first begins to grip. But as far as I know Bilbo alone in history has ever gone beyond playing, and really done it. He needed all my help, too. And even so he would never have just forsaken it, or cast it aside. It was not Gollum, Frodo, but the Ring itself that decided things. The Ring left him.’
The Lord of the Rings Book I, Chapter 2: The Shadow of the Past
It is possible that this is just a figure of speech and the Ring was simply acting under the influence of Sauron (after all, it contains much of Sauron's power). There is certainly a metaphysical question to be solved there, but I'm not sure there is any practical difference.