If you make your chars private no one will see your logs/timers/etc. But the generic MyDDO stuff will be visible.
I remind you all that the info was/his already available and this tools doesn't show anything new that cant be seen already. It just provides a easy way to access that information for those who wish.
For example, I use it in all my chars to check the raid timers and ransacks.
(have 10+ chars, it becomes a mess to follow the timers in all of them without it...)