I am assuming this machine, which I call The Link, is rare. That is, there won't be enough of them running in parallel to improve bandwidth.
I will offer a different view. The Link would not be on a network in the normal sense. There would be no point to it.
First, due to its importance and the low bandwidth, use of The Link would be tightly controlled so people didn't transmit cat pictures. There would be firewalls to prevent unauthorized access.
Second, due to the low bandwidth, The Link can be though of more as a telegraph than something on a modern computer network. A telegraph (barring the need for repeaters) offers speeds comparable to lightspeed thanks to the magic of copper wire. You close the telegraph key, the other end goes "click". Sure the electromagnet is slow, but the human keying the signals is even slower. It is effectively instant. Consider an underwater cable between the US and the UK. Each country might have a sophisticated telegraph network, and for a small fee Sally in Florida can tell grandma in Maine about her new cat, but which messages would be considered for communications over the underwater cable? Probably not the cat telegram. Instead, it would probably be used for information relevant to politics and high finance.
Of course in 2016, we're not going to have a couple of people tapping out messages on our interstellar link. But it's still like a telegraph. You'd have a computer at each end of The Link. The sender would read from a buffer of messages (encoded, then compressed maximally) and tap them out. The machine at the other end would receive, decompress, and decode.
So while there would not be a network protocol, there would probably be some manner of message protocol so the receiver would know when it was appropriate to decompress the message. A short message would be a 'barn burner' to be sure because the compression per character would be smaller and thus less efficient.
Given how controlled the use of The Link would be, it is unlikely that the messages would be particularly interesting to the normal person just the way in our international example above the normal person would not be too concerned about matters of high finance.
But exactly what messages would be sent over The Link?
Say a sub-light colony ship has reached its destination after 300 years and is beginning to build their new home. The Link is set up.
The first messages sent go something like this:
Hello Earth, we have arrived safely and everything is proceeding
according to plan.
(This will be a few characters, perhaps, because of encoding), and answered by,
It's damned good to hear from you, cheerio!
(another 2 or 3 characters)
After pleasantries and diagnostics, what relevance does anything on Earth have to the colony? Help is 300 years away, barring some shocking new discovery. Politics waxes and wanes over the centuries. Countries change. Would the country that sent the ship still exist? Would the World Order that sent the ship be recognizable? What relevance would the colony have to the people of Earth, 15 generations removed from those brave daring souls who boarded the colony ship?
It could be that a cat jpeg might indeed be as useful as any other message.
EDIT - Given the lack of any importance between the day-to-day lives of the people on Earth and the colonists, it would seem The Link in this case would generally be used for low-grade science communications. Observations about the star being orbited, and that sort of thing. I don't know why that would be particularly relevant but it's better than dead air, assuming The Link doesn't wear out from use.
A more likely use of The Link doesn't involved people at all. Instead, the ship housing The Link is purely robotic. These ships are sent by the score to different star systems. They watch, silently and stealthily, for the signals of other races. The data sent back, ever so slowly, is designed to allow humans on Earth some glimpse into the technology of the aliens and hopefully their intent. Sinister, indeed.