They feel if a Death Eater presses it, and it seems they know who.

When another Death Eater presses their own Dark Mark, the other Death Eaters do feel it - both Amycus Carrow and Snape feel their Marks burn when Alecto presses her Dark Mark.

“No she bludgering well won’t!’ bellowed Amycus. ‘Not after the Dark Lord gets hold of her! She’s gorn and sent for him, I felt me Mark burn, and he thinks we’ve got Potter!”
_{^{- Harry Potter and the Deathly Hallows, Chapter 30 (The Sacking of Severus Snape)}}

From this, it seems clear that they know onde the Death Eater who pressed their Dark Mark is - both Snape and Amycus Carrow act certain that the summons came from Alecto inside Hogwarts. Without them actually knowing the location of the Death Eater pressing the Mark, it’d be quite an illogical leap to presume they were within Hogwarts, as there would be several Death Eaters in other locations. Only three were in Hogwarts, so the odds would be against it - they likely did know. It also seems that they can tell que pressed their Dark Mark. Snape seems to know that Alecto was the one who pressed her Mark, despite both her and Amycus being in Hogwarts at the time.

“I was under the impression,’ said Snape, ‘that Alecto had apprehended an intruder.’

‘Really?’ said Professor McGonagall. ‘And what gave you that impression?’

Snape made a slight flexing movement of his left arm, where the Dark Mark was branded into his skin.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 30 (The Sacking of Severus Snape)}}

It is possible that he’d just guessed, but with two Carrows in Hogwarts, it seems less likely that he’d be able to guess correctly without some way of knowing more certainly which one it was.

Os Comensais da Morte do seem to know if the Dark Lord pressed it.

It seems likely that the Death Eaters are able to tell if another Death Eater presses their Dark Mark, or if the Dark Lord has pressed someone’s Dark Mark to summon them all. When the Dark Lord presses Wormtail’s Dark Mark in the graveyard, Death Eaters soon begin Apparating to him there.

“The air was suddenly full of the swishing of cloaks. Between graves, behind the yew tree, in every shadowy space, wizards were Apparating. All of them were hooded and masked.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

The other Death Eaters don’t seem to get summoned when a Death Eater presses their Dark Mark without the Dark Lord being present or trying to summon them to him. When Alecto presses her Dark Mark, the Dark Lord is summoned to her at Hogwarts, but we don’t see any of the other Death Eaters begin to Apparate to Alecto - and Harry was there long enough that if they had he would’ve seen at least some of them, since Apparition is a quick form of transport.

“The sloping-shouldered figure of Alecto Carrow was standing before him, and even as Harry raised his wand, she pressed a stubby forefinger to the skull and snake branded on her forearm.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 29 (The Lost Diadem)}}

The difference between the Dark Lord’s summons and another Death Eater pressing their Dark Mark may possibly be that the Dark Mark only turns black when the Dark Lord himself presses it to summon them to him - it turns black when he uses Wormtail’s to call them to the graveyard.

“He pressed his long, white forefinger to the brand on Wormtail’s arm.

The scar on Harry’s forehead seared with a sharp pain again, and Wormtail let out a fresh howl: Voldemort removed his fingers from Wormtail’s Mark, and Harry saw that it had turned jet black.”
_{^{- Harry Potter and the Goblet of Fire, Chapter 33 (The Death Eaters)}}

When the Death Eaters press their own Dark Mark, though, it doesn’t seem to turn black. Bellatrix presses hers, and it’s not mentioned as turning black, and it isn’t said to when Alecto does, either.

“‘And now,’ she said, in a voice that burst with triumph, ‘we call the Dark Lord!’

And she pushed back her sleeve and touched her forefinger to the Dark Mark.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 23 (Malfoy Manor)}}

It seems like the Dark Mark turning black is the way that the Death Eaters can tell if the Dark Lord is the one who pressed the Dark Mark, so they know to all go to him.

There doesn't seem to be enough information for a definitive answer

No final de cálice de Fogo Snape shows the Dark Mark to Fudge with the following explanation:

Every Death Eater had the sign burned into him by the Dark Lord. It was a means of distinguishing one another, and his means of summoning us to him. When he touched the Mark of any Death Eater, we were to Disapparate, and Apparate, instantly, at his side.

Now, when the Dark Mark is pressed there are two possibilities as to who pressed it. Either it was pressed by Voldemort or by someone other than Voldemort. It is obviously possível for a non-Voldemort to press it (i.e. no protective enchantments that prevent everyone else from pressing it), as throughout Relíquias da Morte Death Eaters press it or talk about pressing it.

If the mark could have been pressed either by Voldemort or non-Voldemort then we have to address how the Death Eaters were to Apparate to him when he presses it (i.e. how they knew that it was Voldemort summoning them and not some random Death Eater pressing it for some other reason). There are two basic possibilities here. Either there is a way to distinguish whether the mark was pressed by Voldemort or non-Voldemort, or no non-Voldemort would ever press the mark.

The second option has a couple of weaknesses. First, as already mentioned, throughout Relíquias da Morte we see that it is clearly not true that no non-Voldemort would ever press it. Of course, it is possible that Voldemort changed his policies and now allowed others to press the mark as well, but that would seem unlikely. The "everyone Apparate when you feel the mark" rule was the longstanding policy from at least before Voldemort originally lost his body (as the Death Eaters all knew to Apparate to the graveyard). Additionally, if Voldemort did change the policy then he would now need a new way to summon the Death Eaters to him, since they would have no way of knowing whether it was pressed by him or by someone else. And furthermore, if there was no way to distinguish whether Voldemort or non-Voldemort pressed the mark, it would be easy for someone to set a trap if they captured one Death Eater. (However, this doesn't necessarily prove anything because it's possible that no one on either side thought of this, and anyway even within the other possibility it would be possible to set such a trap for Voldemort himself.)

This leaves us with the first possibility, where the Death Eaters know whether it was Voldemort or non-Voldemort that pressed the mark, and they accordingly know whether to Apparate or not.

The question, then, is whether the origin of the mark can be further distinguished beyond Voldemort/non-Voldemort to which Death Eater actually originated it. This question can perhaps be further subdivided into whether only Voldemort can distinguish it, or any Death Eater can distinguish it.

There doesn't seem to be any clear evidence that Voldemort can distinguish the originator of the mark. When he is attending to Grindelwald we have the following description of his reaction to the mark being pressed:

he was enraged at the summons he felt — he had warned them, he had told them to summon him for nothing less than Potter. If they were mistaken...

And shortly thereafter:

They would suffer his retribution if they had no good reason for calling him back.

There is nothing there to indicate that he knew that it was Bellatrix that had summoned him. If anything, from the vague "they" and "them" it would seem that he did not know, though one could argue that he was just generally referring to anyone who was involved or who could have summoned him.

The Death Eaters do vie for the honor of pressing it, though, which could indicate that Voldemort could distinguish its origin. However, it is also possible that he could find out who pressed it via other means (e.g. politely asking, torture, Legilimency, etc.).

On the other hand, when Alecto summons Voldemort, both Amycus and Snape appear to know that she was the one who pressed the mark:

"No she bludgering well won't!" bellowed Amycus. "Not after the Dark Lord gets hold of her! She's gorn and sent for him, I felt me mark burn, and he thinks we've got Potter!"

"I was under the impression," said Snape, "that Alecto had apprehended an intruder."

"Really?" said Professor McGonagall. "And what gave you that impression?"

Snape made a slight flexing movement of his left arm, where the Dark Mark was branded into his skin.

"Oh, but naturally," said Professor McGonagall. "You Death Eaters have your own private means of communication, I forgot."

This could indicate that the Death Eaters pode distinguish who sent the summons, and if they can then a fortiori Voldemort can.

However, it is also possible that Amycus and Snape simply guessed that it was Alecto that had pressed the mark. This is all the more possible when we consider that even if the originator cannot be distinguished, the person's location can be¹ or else a summons would never be able to work in the first place. If so, it is possible that they simply knew that the summons was to Hogwarts, in which case it would be pretty obvious to them that it was Alecto who had pressed it (as the plan had been orchestrated with Alecto as the one waiting for Harry in Ravenclaw Tower).

In conclusion, I think it is likely that everyone can distinguish between a Voldemort summons and a non-Voldemort summons, but it is not quite clear whether anyone can distinguish between different non-Voldemort summonses.

^{1. Though one could argue that they don't actually know the location but merely Apparate and are somehow guided there.}