The messageboad (and since October) the website is database driven. That means, all content is retrieved from a database and displayed through a template system. This is why you can change the layout of the website for you.
Having said that, if you have physical access to the database, you are able to read everything which is stored in there. I have physical access to the database and I know the access passwords. I can backup and restore the database. Hence I can access private messages too.
The forum software however, has no admin interface for private messages. You can check this at phpbb.com or just ask some forum members who are running a phpbb themselves. I cannot see PMs with just a few mouse clicks. I have to go through the backup and restore process and then manually query the restored database.
In a nutshell, I can access them if there would be a need for it, but this is not a 5 minute and 2 mouse-clicks job.
And fyi, this is practially the same with your emails stored in your providers mailserver. They could read it if there would be a need for that.
Hope that answers the questions.
Edit: Spelling. Do we need a spellchecker here?
