they are either from Mary or the Musgroves. The text reads:

The first three or four days passed most quietly, with no circumstance to mark them excepting the receipt of a note or two from Lyme, which found their way to Anne, she could not tell how, and brought a rather improving account of Louisa. IMO she never questions where the notes come from, just how they found their way. Getting a letter took much longer in those days and she received the news almost immediately after the fact. I cant imagine CW would have left the notes unsigned if he'd written them himself. I guess, he must have mentioned the visit to his sister in Lyme and someone then gave him the letter to drop of at Anne's.