you should be able to view the particulars of the file at your local authority offices, if not on-line.
your neighbour is allowed build on a common wall (party wall) once theres no encroaching in adjoining land by any construction. this isnt generally done without neighbouring permission because eaves overhangs and parapets etc tend to encroach onto adjoining land.
if you are really worried about this you should hire a professional (architect / engineer) to prepare an objection, or at least advise you.
are you saying the site notice was taken down? perhaps the application wasnt valid.. again, check with your LA.