Steven Zahm@ Bryan
There is a settings to require users to login before the directory can be viewed. You can double this up by also publish the page the directory is embedded on as Private instead of Public (a core WP feature)
Many of the fieldsets in Connections such as address, phone and email can be marked as private or public. Assuming the directory does not require a login to view, the items set as public would be displayed while the private would require a user to be logged in.
The login is handled by WP, not Connections. So, you would need to use core WP feature to manage the WP users in regards to string passwords and password reset. Or use any of the popular plugins available which enhance those features.
Hope this answers your questions thoroughly!