Not sure of any magazines, though we have obviously the Southwark News and South London Press, the latter probably only about 30% relating to Southwark. The Southwark News is very good.
I like reading both, but you'll probably hear about most things in them on this site first, sometimes weeks before. The editor, James, does an excellent job. The news page is on the "home page" - http://www.london-se1.co.uk/.