There is a .dropbox folder in the HOME folder (whatever that is set to). It contains the following files:
$ file *
config.db: SQLite 3.x database
dropbox.pid: ASCII text, with no line terminators
host.db: ASCII text
host.dbx: ASCII text
photo.dbx: SysEx File -
config.db doesn't contain any interesting information, the directory "l" is empty, and I can't easily get any information out of the other files. It doesn't seem unlikely that all preferences are stored here, though.
Still, running both instances from a non-default HOME seems like sound advice.