The fact that they wrote instead of saying it direct to each other shows how much away the were from one another.
the second alliteration (for he went west) shows that the man didnt have any assitation to leave his ex lover so he made the right decision and he wet to west.
On the contrary the woman decided to go to east, and in my opinion there is no alliteration in this point, even though is quite easy to understand that it wasnt easy for the woman to take the decision to leave him and to go on with her life.
until the last verse, there is the word I two times in the same W - question (the personal pronom change sowohl...) and in the 5ten verse you can find two times the word went, from wich you understand that the language that the poet uses is everyday english (i apopsi mou einai oti olo to poihma einai se everyday english)