The letter from a friend doesn't appear until the completion of dragon rising. This mean's the "friend" only believes you're dovahkiin after you've met the greybeards, so then it is my belief that one of the greybeards is sending these letters based on
A) The Greybeards are the one of the few groups of people capable of hearing the Dovahkiin's shout anywhere across Tamriel. (they heard it easily enough from western watch)
B) They would be most knowledgable of the wherabouts of word walls, as they can hear the resonating echoes of the walls (as shown by arngeir's ability to tell you wear they are. "we have heard a whisper...")
the interest point is that Arngeir is probably not the one sending the letters, I believe this based on the fact that he wants to temper your training to a certain speed and reveals the locations of word walls whenever you've found a previous one. the "friend" similarly will not reveal a new location until you have found the last one, but for arngeir to be revealing two locations at a time seems a bit strange. what do you think, who is the Dovahkiin's mysterious "friend"?
~I have considered Delphine as she signs her note "A friend" after having stolen the horn of jurgen windcaller, and is reasonably paranoid enough of her note being discovered by thalmor, and wouldn't want her location tortured out of a courier if her name was on it. she is extensive in her research but she would not have known whon the dovahkiin was before the completion of "the horn of jurgen windcaller" and she would most likely not know the exact location of word walls