If you've set the private forum's Posting Restrictions
to Mods & Admins the easiest way to accomplish having your mods be able to access and there is to assign them as moderators of the staff forum which will give them access to it. You have to assign them as a moderator there for them to be able to enter and post inside if you have that Posting Restriction
set to Mods & Admins.
If you like you can choose to set the Posting Restrictions
to Specified Users, Mods & Admins
and then you can select the mods names from your userlist and on the select menu pull down to Assign Private Forum Access.
When you have a private forum hidden
you have to take this a step further and select them from the userlist and on the menu go down to Assign Hidden Forum Viewing Ability
which also applies if you have forums you want some users to be able to see and others not to.
If you use MBA on a forum it will send admin's an email with the person's request for entry and you can accept or reject their request.