Yes. Here is what Tolkien said:
But to achieve this he had been obliged to let a great part of his own inherent power (a frequent and very significant motive in myth and fairy-story) pass into the One Ring. While he wore it, his power on earth was actually enhanced. But even if he did not wear it, that power existed and was in 'rapport' with himself: he was not 'diminished'. Unless some other seized it and became possessed of it. If that happened, the new possessor could (if sufficiently strong and heroic by nature) challenge Sauron, become master of all that he had learned or done since the making of the One Ring, and so overthrow him and usurp his place.
The reason we don't see the effects of the Ring all the time is threefold:
- Because wielding the Ring takes effort and practice. "Before you could use that power you would need to become far stronger, and to train your will to the domination of others", in the words of Galadriel. In fact, she says that without this training, the Ring would destroy Frodo.
- You have to actually claim the Ring as your own to truly use its power. Frodo and Bilbo never did this, and maybe Gollum just never had the strength. The moment Frodo claimed the Ring for his own, Sauron knew. "And far away, even as Frodo put on the Ring and claimed it for his own, even in Sammath Naur the very heart of his realm, the Power in Barad-Dur was shaken, and the Tower trembled from its foundations to its proud and bitter crown. The Dark Lord was suddenly aware of him[...]
- The power of the Ring is not absolute. Sauron was able to dominate and subvert the Numenoreans only with time and subtlety. And the reason Sauron had to do that? "So great was the might and splendour of the Númenóreans that Sauron’s own servants deserted him." So even Sauron wielding the Ring wasn't able to control his own servants under some circumstances.
Bilbo and Gollum are actually good examples of the power of the Ring: forcing a mortal to stay alive is tremendous. The Ring doesn't grant you courage, strength, wisdom, or magic spells. It gives you the power to dominate others, strengthens your native ability to read minds, and, if you're a mortal, forces you to stay alive - not even the Ring can fundamentally change fate decreed by Illuvatar; the Ring is already stretching The Rules quite a bit here.
The Ring granted Frodo and Sam it's chief power - domination over others - on several occasions, to subdue Gollum and to terrify Orcs. This was before Frodo even claimed the Ring.
Then suddenly, as before under the eaves of the Emyn Muil, Sam saw these two rivals with other vision. A crouching shape, scarcely more than the shadow of a living thing, a creature now wholly ruined and defeated, yet filled with a hideous lust and rage; and before it stood stern, untouchable now by pity, a figure robed in white, but at its breast it held a wheel of fire. Out of the fire there spoke a commanding voice.
‘Begone, and trouble me no more! If you touch me ever again, you shall be cast yourself into the Fire of Doom.’
The crouching shape backed away, terror in its blinking eyes, and yet at the same time insatiable desire.
Then the vision passed and Sam saw Frodo standing, hand on breast, his breath coming in great gasps, and Gollum at his feet, resting on his knees with his wide-splayed hands upon the ground.
For what it saw was not a short frightened hobbit trying to hold a steady sword: it saw a great silent shape, cloaked in a grey shadow, looming against the wavering light behind; in one hand it held a sword, the very light of which was a bitter pain, the other was clutched at its breast, but held concealed some nameless menace of power and doom. For a moment the Orc crouched, and then with a hideous yelp of fear it turned and fled back as it had come.
We also have some examples of what would have happened, from Tolkien's letters. For example, what if Gollum hadn't been there at Mount Doom?
Frodo too would then probably, if not attacked, have had to take the same way: cast himself with the Ring into the abyss. If not he would of course have completely failed. It is an interesting problem: how Sauron would have acted or the claimant have resisted. Sauron sent at once the Ringwraiths. They were naturally fully instructed, and in no way deceived as to the real lordship of the Ring. The wearer would not be invisible to them, but the reverse; and the more vulnerable to their weapons. But the situation was now different to that under Weathertop, where Frodo acted merely in fear and wished only to use (in vain) the Ring's subsidiary power of conferring invisibility. He had grown since then. Would they have been immune from its power if he claimed it as an instrument of command and domination?
Not wholly. I do not think they could have attacked him with violence, nor laid hold upon him or taken him captive; they would have obeyed or feigned to obey any minor commands of his that did not interfere with their errand – laid upon them by Sauron, who still through their nine rings (which he held) had primary control of their wills. That errand was to remove Frodo from the Crack. Once he lost the power or opportunity to destroy the Ring, the end could not be in doubt – saving help from outside, which was hardly even remotely possible.
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. Even so for a long time his acts and commands would still have to seem 'good' to him, to be for the benefit of others beside himself.
It also granted Sam the ability to understand the speech of Orcs, and Frodo was able to understand Galadriel's thoughts: "[Frodo] perceived my thought more clearly than many who are accounted wise."
However, in the end, the Ring - as essentially distilled Sauron and "power" - would have turned anyone who used it "evil." From Tolkien's letters:
In any case Elrond or Galadriel would have proceeded in the policy now adopted by Sauron: they would have built up an empire with great and absolutely subservient generals and armies and engines of war, until they could challenge Sauron and destroy him by force.
Confrontation of Sauron alone, unaided, self to self was not contemplated. One can imagine the scene in which Gandalf, say, was placed in such a position. It would be a delicate balance. On one side the true allegiance of the Ring to Sauron; on the other superior strength because Sauron was not actually in possession, and perhaps also because he was weakened by long corruption and expenditure of will in dominating inferiors. If Gandalf proved the victor, the result would have been for Sauron the same as the destruction of the Ring; for him it would have been destroyed, taken from him for ever. But the Ring and all its works would have endured. It would have been the master in the end.
Gandalf as Ring-Lord would have been far worse than Sauron. He would have remained ‘righteous’, but self-righteous. He would have continued to rule and order things for ‘good’, and the benefit of his subjects according to his wisdom (which was and would have remained great)."
EDIT: To clarify, the One Ring did enhance Sauron's powers as well, and it was not created solely to control the wielders of the other Great Rings.
From Tolkien's letters, the Ring was created in "his [Sauron's] effort (largely unsuccessful) to enslave the Elves, and in his desire to establish a control over the minds and wills of his servants."
Additionally, we know Sauron used the One Ring to subjugate the Numenoreans. From Letter 211:
He naturally had the One Ring, and so very soon dominated the minds and wills of most of the Númenóreans.
Thus, the Ring could be used by Sauron used to dominate the minds and wills of people who were not wearing other rings. Wielding another one of the Great Rings only seems to have made you especially vulnerable.
The reason we didn't see any amazing tricks from Frodo or Sam (beyond terrifying and subjugating Gollum, Orcs, etc.) is explained in the book, and in one of my quotes above. Frodo asks Galadriel why he can't use the Ring to know the thoughts of the wielders of the Three:
'Only thrice have you set the Ring upon your finger since you knew what you possessed. Do no try! It would destroy you. Did not Gandalf tell you the rings give power according to the measure of their possessor? Before you could use that power you would need to become far stronger, and train your will to the domination of others.'
Using the power of the Ring is not as simple as "put it on, get amazing results."