Maybe i am wrong about what is a private wiki, but for me, a private wiki is a wiki that require authentication to see pages. What we do to make a private wiki is remove permission to read ans edit page to all user and give those permission to know user :
$wgGroupPermissions['*']['createaccount'] = false;
$wgGroupPermissions['*']['read'] = false;
$wgGroupPermissions['*']['edit'] = false;
$wgGroupPermissions['*']['createpage'] = false;
$wgGroupPermissions['*']['createtalk'] = false;
$wgGroupPermissions['*']['writeapi'] = false;
$wgGroupPermissions['user']['edit'] = true;
$wgGroupPermissions['user']['createpage'] = true;
$wgGroupPermissions['user']['createtalk'] = true;
If i remove those line in my LocalSettings file, it is a public wiki and anyone can read or edit pages.
But like i said, maybe i am wrong about what is a private wiki.