In phpbb can moderators read private messages sent between members or are they safe from prying eyes?

Answer 1

You need to have access to the SQL database.

If so, it is a simple matter of downloading the "messages" table and browsing it with some hex editing software.

Answer 2

If the owner of the account has given them that privledge than yes they can. I would be carefull of what you say even though you are sending a PM(private message), it just may be read by more than you want.