Horcruxes all have a degree of autonomy.
All Horcruxes have the ability to possess others who get emotionally close to them. Though the diary was the only soul fragment that we see gain a somewhat physical form, all Horcruxes have that ability, so it’s possible that the others could have taken a physical form if given that same opportunity.
“Hang on,’ said Ron, frowning. ‘The bit of soul in that diary was possessing Ginny, wasn’t it? How does that work, then?’
‘While the magical container is still intact, the bit of soul inside it can flit in and out of someone if they get too close to the object. I don’t mean holding it for too long, it’s nothing to do with touching it,’ she added, before Ron could speak. ‘I mean close emotionally. Ginny poured her heart out into that diary, she made herself incredibly vulnerable. You’re in trouble if you get too fond of or dependent on the Horcrux.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
The locket also seems to get close to possessing Ron, since Ron’s eyes turn red after listening to the illusions the locket projected and having them emotionally affect him, though in his case it was a negative effect on his emotions that allowed it, not a positive one like Ginny’s ‘friendship’.
“Who wouldn’t prefer him, what woman would take you? You are nothing, nothing, nothing to him,’ crooned Riddle-Hermione, and she stretched like a snake and entwined herself around Riddle-Harry, wrapping him in a close embrace: their lips met.
On the ground in front of them, Ron’s face filled with anguish: he raised the sword high, his arms shaking.
‘Do it, Ron!’ Harry yelled.
Ron looked towards him and Harry thought he saw a trace of scarlet in his eyes.”
- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)
Therefore, it seems logical that it’s an ability all Horcruxes have, but most of the Dark Lord’s Horcruxes were destroyed without them having an opportunity to use that ability, because no one got emotionally close enough to them that they could.
They likely don’t know each other’s status.
The Horcruxes don’t seem to have any conscious knowledge of the others. The Dark Lord doesn’t know through Nagini, one of his Horcruxes, that any of the other Horcruxes were destroyed. If Horcruxes were innately aware of each other’s status, then presumably Nagini would have informed him of their destruction. Instead, Nagini goes with him to check on them.
“He would visit the Gaunt shack first, then, and take Nagini with him: he would not be parted from the snake any more … And he strode from the room, through the hall and out into the dark garden where the fountain played; he called the snake in Parseltongue and it slithered out to join him like a long shadow …”
- Harry Potter and the Deathly Hallows, Chapter 27 (The Final Hiding Place)
This indicates that the Horcruxes aren’t aware of the status of the others, since Nagini could communicate with the Dark Lord and would have informed him if she knew others were being destroyed. Therefore, they also wouldn’t be aware if any others gain physical form, and would be unable to prevent two soul fragments having a physical form at the same time.
But, they may sense others’ presence.
Though the Horcruxes aren’t consciously aware of the others, they may be able to sense another’s presence, though it’s unclear if they’re aware what they’re sensing. When Nagini in Bathilda’s corpse is around Harry, the Horcrux on his chest reacts. However, it’s not clear if the reaction is because of proximity to another Horcrux or something else entirely, like Harry’s concern and fear.
“Miss Bagshot?’ Harry repeated, and he advanced, with the picture in his hands, as the flames burst into life in the fireplace. Bathilda looked up at his voice and the Horcrux beat faster upon his chest.”
- Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda’s Secret)
Later, when Nagini summons the Dark Lord, the Horcrux on his chest twitches, which may be a reaction to that.
“Then she closed her eyes and several things happened at once: Harry’s scar prickled painfully; the Horcrux twitched so that the front of his sweater actually moved; the dark, fetid room dissolved momentarily. He felt a leap of joy and spoke in a high, cold voice: hold him!”
- Harry Potter and the Deathly Hallows, Chapter 17 (Bathilda’s Secret)
It seems possible from this, but isn’t made clear enough to be certain, that Horcruxes can ‘sense’ the presence of another Horcrux, though it may be an instinctive pull towards a piece of the same soul rather than a conscious awareness if there is a reaction at all.
Horcruxes are aware of some things.
Horcruxes do have a certain level of awareness. They can detect whether someone is an enemy, and act against them. The locket tried to choke Harry while he was underwater.
“It was not weed: the chain of the Horcrux had tightened and was slowly constricting his wind pipe.
Harry kicked out wildly, trying to push himself back to the surface, but merely propelled himself into the rocky side of the pool.”
- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)
From that example, they’re able to attempt to actively destroy their enemies to a certain extent. Harry suspected that the Horcrux sensed the sword’s presence, and tried to kill him so that he wouldn’t be able to get it from the lake.
“The Horcrux was still swinging from Ron’s hand. The locket was twitching slightly. Harry knew that the thing inside it was agitated again. It had sensed the presence of the sword and had tried to kill Harry rather than let him possess it.”
- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)
Also, they’re able to sense how best to affect certain people. The diary manipulated Ginny into befriending it, and the locket shows this same ability. It plays on Ron’s fears and uses them to draw emotion out of him.
“Then a voice hissed from out of the Horcrux.
‘I have seen your heart, and it is mine.’
‘Don’t listen to it!’ Harry said harshly. ‘Stab it!’
‘I have seen your dreams, Ronald Weasley, and I have seen your fears. All you desire is possible, but all that you dread is also possible …”
- Harry Potter and the Deathly Hallows, Chapter 19 (The Silver Doe)
Therefore, Horcruxes do have a certain level of awareness. They’re not aware of the others’ status, but they are aware of other things, and use this to their advantage.
Horcruxes mightn’t be able to get full form.
It’s never mentioned, nor is it logical to conclude from what’s known, that the soul fragment in any Horcrux (including the diary) can become a fully functional Dark Lord if it gains enough strength. Secrets of the Darkest Art explains that Horcruxes can possess others if they become emotionally close enough, but it never mentions the soul fragment contained in a Horcrux being able to form a fully functional body and becoming like another of the wizard who created it, which it’d almost certainly mention if this was also a known effect of the soul fragment gaining enough strength.
“Hang on,’ said Ron, frowning. ‘The bit of soul in that diary was possessing Ginny, wasn’t it? How does that work, then?’
‘While the magical container is still intact, the bit of soul inside it can flit in and out of someone if they get too close to the object. I don’t mean holding it for too long, it’s nothing to do with touching it,’ she added, before Ron could speak. ‘I mean close emotionally. Ginny poured her heart out into that diary, she made herself incredibly vulnerable. You’re in trouble if you get too fond of or dependent on the Horcrux.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
Despite their abilities, Horcruxes are still only soul fragments dependent on their containers for survival.
“But my point is that whatever happens to your body, your soul will survive, untouched,’ said Hermione. ‘But it’s the other way round with a Horcrux. The fragment of soul inside it depends on its container, its enchanted body, for survival. It can’t exist without it.”
- Harry Potter and the Deathly Hallows, Chapter 6 (The Ghoul in Pyjamas)
Though the soul fragments clearly can gain more strength, it’s unclear if this strength is enough to make them fully corporeal. The young Tom Riddle in the diary is certainly more powerful than a ghost, since he can use a wand, but it’s not clear that he’s actually a fully corporeal being, or that taking all of Ginny’s life force would make him one.
“Riddle was pointing Harry’s wand at Fawkes; there was a bang like a gun and Fawkes took flight again in a whirl of gold and scarlet.”
- Harry Potter and the Chamber of Secrets, Chapter 17 (The Heir of Slytherin)
Though it’s true he grows more solid and capable of physical interaction with his surroundings as he gains power, it’s never made clear what he’d become if he gained all the strength he possibly could. It’s never said that he’d become a fully corporeal Dark Lord himself, so there’s no basis to conclude that he would.
