Thanks to /r/historicusXIII for inviting me to the thread.

When looking at the three letters of a ship's callsign in 1912, the first letter related to the wireless company that owned and operated the equipment: so the M of Titanic's callsign MGY stood for 'Marconi'. Callsigns were allocated by the wireless companies themselves and, of course, the vast majority of ships in 1912 with wireless stations on board were manned by Marconi operators (just look at how many of the ships answering Titanic's distress call had callsigns beginning with M).

Given the above, I would assume that the Marconi company would have distributed books to their operators that contained all the allocated ship and shore-based Marconi Station callsigns. I can't evidence this yet (I'll do some more digging tomorrow) but it seems a logical assumption. Therefore, it is likely that the ships hearing Titanic's CQD/SOS knew who she was.

Other factors that might be relevant include the fact that the Marconi operators all knew each other (for the most part) and could recognise an individual by the sound of their 'keying'. Similarly, individual wireless stations had different sounding 'sparks' that were recognisable. Therefore, even without hearing a callsign, a message source might have been recognised by a listener.

The expert on this sort of thing, and who would definitely know the answer, is a chap called Parks Stephenson, who worked with Cameron on his Titanic film. Perhaps he is lurking..?

EDIT: Apparently, the ship's log on board Carpathia was annotated: "Heard the Titanic invoke CQD and SOS" - so Carpathia definitely knew which ship was sinking.

I've looked for an answer, and can't find anything to satisfy your question. We do know that Carpathia knew MGY was Titanic, but how they knew it is a mystery to me. Maybe /u/darkninjapenguin or /u/unclearthur know more about this.