The workings of the fake Galleon communication system is described very specifically in Order of the Phoenix:
Hermione soon devised a very clever method of communicating the time and date of the next meeting to all the members in case they needed to change it at short notice, because it would look so suspicious if people from different Houses were seen crossing the Great Hall to talk to each other too often. She gave each of the members of the D.A. a fake Galleon (Ron became very excited when he saw the basket at first, convinced that she was actually giving out gold).
"You see the numerals around the edge of the coins?" Hermione said, holding one up for examination at the end of their fourth meeting. The coin gleamed fat and yellow in the light from the torches. “On real Galleons that’s just a serial number referring to the goblin who cast the coin. On these fake coins, though, the numbers will change to reflect the time and date of the next meeting. The coins will grow hot when the date changes, so if you're carrying them in a pocket you’ll be able to feel them. We take one each, and when Harry sets the date of the next meeting he'll change the numbers on his coin, and because I’ve put a Protean Charm on them, they’ll all change to mimic his."
This appears to be a very simple system. Each individual coin is charmed to duplicate Harry's coin. So whatever appears on Harry's coin will appear on everyone else's coin as well.
Yet later in the series we see messages being passed from one coin to another without the use of Harry's coin. At the end of Half-Blood Prince Harry tells Ron and Hermione to rouse the DA members via the coins, yet he does not give them his coin nor does he tell them where his coin is so that they could use it:
Use anyone else who you can rustle up from the D.A., Hermione, those contact Galleons will still work, right?
If Hermione changed the numbers on her own coin, it should not have transmitted any message to anyone else's coin because their coins were duplicates of Harry's coin, not hers.
Similarly, in Deathly Hallows we find Neville and Luna both using their coins to send messages:
"Neville, she's all right, we've seen her —"
"Yeah, I know, she managed to get a message to me."
From his pocket he pulled out a golden coin, and Harry recognized it as one of the fake Galleons that Dumbledore's Army had used to send one another messages.
"These have been great," said Neville, beaming at Hermione. "The Carrows never rumbled how we were communicating it drove them mad.
Note that the above quote apparently claims that the D.A. had used the coins to send one another messages, something that was never mentioned in Order of the Phoenix and is certainly beyond the scope of the system Hermione described.
Also in Deathly Hallows:
"I got the message," she said, holding up her own fake Galleon, and she walked over to sit beside Michael Corner.
"We're fighting, aren't we?" said Dean, taking out his fake Galleon. "The message said Harry was back, and we were going to fight! I'll have to get a wand, though —"
Once again, these are communications that were made without the use of Harry's coin.
So when and how exactly did the communication system change? How did it go from a bunch of coins being charmed to duplicate whatever message was written on Harry's coin, to the ability for anyone to communicate with anyone?