I have always been puzzled that - from Moria, to Lothlorien under under its eaves, then down the River Anduin, Frodo never thought to mention to other members of the Fellowship, that he could hear the sounds of someone following - and that he suspected it was Gollum.
(Page numbers are from the Kindle edition of Lord of the Rings):
p311: Yet Frodo began to hear, or to imagine that he heard, something else: like the fall of soft bare feet. It was never loud enough, or near enough, for him to feel certain that he heard it; but once it had started it never stopped, while the company was moving. But it was not an echo, for when they halted it pattered on for a little all by itself; and then grew still.
OK; Frodo wasn't 'certain' - but why not mention it?
Just to be on the safe side, and to alert the others?
p314: Frodo's spirits rose a little, but he still felt oppressed, and still at time he heard, or thought he heard, away behind the company and beyond the fall and patter of their feet, a following footstep that was not an echo.
p317: [Frodo's] watch was nearly over when, far off where he guessed that the western archway stood, he fancied that he could see two pale points of light, almost like luminous eyes.
Frodo manages to convince himself he was dreaming the eyes, but given the previous twice times he 'thought' he had heard footsteps, and what he knew of Gollum from Bilbo; surely now would be the time to voice his suspicions and put the Fellowship onto alert.
In between escaping from Moria, and reaching the woods of Lothlorien, Frodo is at the rear with Gimli who says he can hear nothing. But then, hobbits hear better than dwarves.
p337: Yet [Frodo] had heard something, or thought he had. As soon as the shadows had fallen about them, and the road behind was dim, he had heard again the quick patter of feet. Even now he heard it. He turned swiftly, There were two tiny gleams of light behind, or for a moment he thought he saw then, but at once they slipped aside and vanished.
'What is it?' said the dwarf.
I don't know', answered Frodo. I thought I heard feet, and a thought I saw a light - like eyes. I have thought so often, since we first entered Moria.'
At last, Frodo has mentioned it! But when Gimli hears nothing by lying with his ear to the ground (!) - this supposedly settles the matter negatively, and no more is said or done.
That same night, Frodo actually sees Gollum's face, after Gollum has climbed up to the tree platform where the hobbits slept.
But Gollum escapes when Haldir the elf returns. Haldir also sees something not an orc, that he thought was like a hobbit, except for being skilled in trees (so Frodo knows for sure he did not imagine it). But the mystery climber is never mentioned to the company, nor his identity speculated upon.
By now, Frodo has surely put two and two together and knows that they are being followed by Gollum; but still says nothing to the company at large.
Only when, after leaving Lorien, when they are travelling in boats down the Anduin, and Sam reports seeing 'a log with eyes' that is catching up with the boats, and 'puts a name' to this creature; does Frodo actually discuss his previous observations of Gollum, and reveal - but only to Sam! - that he had already noticed that something was trailing the company, and who it was.
Later that night, after Gollum has attempted an attack and Aragorn has been roused, it turns-out that Aragorn has also known that Gollum has been following them "all through Moria and right down to Nimrodel".
I must say; I find this kind of secrecy inexplicable!
If I had been another member of the Fellowship, I would certainly have appreciated being told, whether by Frodo or Aragorn (who became their leader after Gandalf's fall) that we has Gollum on our trail.
Not least, those of the company who were standing watch, ought to have been told what Gollum looked like, and his capabilities, so that they would know the kind of threat to look-out-for.
In fact; if I had been Boromir, Legolas, Gimli, or one of the other hobbits; I would have been pretty angry if I discovered that such a vital piece of information was known but being withheld (independently of each other) for many days, both by Frodo and Aragorn.
My assumption is that Tolkien wanted to spin-out the suspense, and not to name Gollum - but instead to let the reader piece-together the clues; and only gradually realize that someone was following the Fellowship, and whom.
But I think he overdid it!
To the point of generating pointlessly, and indeed dangerously, secretive behaviour that is implausible from Frodo, and even more so from Aragorn.