I ask this mainly because mortals turn invisible when they use any ring of power, also Frodo in the two towers book says to Smeagol something along the lines of
"If I were to put the ring on you would obey my command "
But if Frodo was invisible, wouldn't it be a bit hard to command something as it won't be able to see you?
The One Ring only bestows power according to the measure of its wearer. The One had given Gollum, for example,
power according to his stature [FotR]
Galadriel confirms this with Frodo in Lorien,
"why cannot I see all the others and know the thoughts of those that wear them?"
"You have not tried. 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." [FotR]
Gandalf did in fact tell him. Elrond too speaks of those who would wield it;
Its strength, Boromir, is too great for anyone to wield at will, save those who have already a great power of their own. [FotR]
To exercise the Ring's power one would have to look toward domination, have a strong will, and the power wielded is only measured by the stature of the wearer. Saruman, Gandalf, Galadriel could use its full might and they'd have a chance to face and topple Sauron personally with the One. Someone like Frodo
would have been utterly overthrown: crushed to dust, or preserved in torment as a gibbering slave. [The Letters of J.R.R. Tolkien]
So yes, donning the Ring, after much time training Frodo could start to dominate the minds of others. Domination of the minds of others is a feature of the One. Note Sauron's use of it in Numenor.
Highly theoretically, it has to do with will; as Galadriel says to Frodo:
'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 them?'
'You have not tried,' [Galadriel] 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.
Fellowship of the Ring Book II Chapter 7: "The Mirror of Galadriel"
Even ignoring the fact that Frodo can still speak whilst invisible, the Ring appears to be controlled not as much by vocal commands but by the desire of the wielder.
Tolkien discusses the possibility of Frodo using the Ring in Letter 246, and it basically comes down to strength of will:
Frodo had become a considerable person, but of a special kind: in spiritual enlargement rather than in increase of physical or mental power; his will was much stronger than it had been, but so far it had been exercised in resisting not using the Ring and with the object of destroying it. He needed time, much time, before he could control the Ring or (which in such a case is the same) before it could control him; before his will and arrogance could grow to a stature in which he could dominate other major hostile wills.
The Letters of J.R.R. Tolkien 246: To Mrs. Eileen Elgar (draft). September 1963
In this particular case, though I don't believe Frodo is being entirely truthful; discussing the power of the Ring, Tolkien writes:
[T]he Ruling Ring [...] contained the powers of all the others, and controlled them, so that its wearer could see the thoughts of all those that used the lesser rings, could govern all that they did, and in the end could utterly enslave them.
The Letters of J.R.R. Tolkien 131: To Milton Waldman. 1951
Which suggests that the One can control the bearer of the Seven, the Nine, and the Three, but cannot inherently dominate the minds of individuals not wearing Great Rings. In the specific case mentioned in the question, it seems more likely that Frodo is simply trying to intimidate Gollum, rather than making an actionable threat.
My feeling, when I did read it a long time ago, was, that
the Ring had something like its own kind of will (the Ring left Gollum, betrayed another his owner by falling down in the worst moment and so) - not exactly human inteligence and will, but more like good trained dog - usually it followed commands of its current owners, but it could oppose them to some small amount and could lure them to be used more and more and finally could takeover on them. Gandalf also said, that Bilbo lasted so long, because he started the ownership with (sorry, have not the word) good intentions and spared Gollum, and also because he did not use it so much and often
Gollum was with the Ring long time, was highly modified by it (started as regular hobbit, ended like creature) and was really bound to it (my precious). He was under the power of the Ring even when he no longer owned it. And the Ring refused to be owned by Gollum at the end (on the way there and back). So if the Ring accepted Bilbo as its owner, Gollum was kind of bounded by it too. Not happy, not willingly, but still.
and Gollum promised by the Ring to obey Frodo, who was next in ownership line and was there lawfully. So he visibly respected the "will" or "choice" of the Ring (to some limits). And Gollum was not truly and entirely evil - he was afraid to break the old rules of the Game of Questions - at least openly and without magic power on his side. So there was something small even inside Gollum, what was "good" and sometimes modified his actions
so Frodo, as lawful owner of the Ring and its current respected ruler, had some small power over Gollum, the slave of the Ring. Also Frodo was young and energetic and a "brave hero", Gollum was an old, broken "poor creature" even in his own mind and so could not so simply openly resist the full power of a command from the Ring's owner. (But still would plan to betray him later, when time is right and chances are good and so on and so on ...)
