There doesn't seem to be enough information for a definitive answer
At the end of Goblet of Fire 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 possible for a non-Voldemort to press it (i.e. no protective enchantments that prevent everyone else from pressing it), as throughout Deathly Hallows 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 Deathly Hallows 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 can 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.
