Is it possible to make registration for Active Social private?  I don't mean verified, I mean private, as in I have to manually approve new members before they can access the Active Social modules.

I would prefer to do this using the AS registration module rather than Dynamic Registration.

You could manage this through core DNN security. Just make sure you pages/as modules are not set for Registered Users..instead set them up for something like "Approved Members" or "Customers"....then when you manually add them to the role "Approved Members" they can now access the modules.

I'm in the process of setting up a corporate site this way as a way for our customers to communicate to our account managers (and other customers) through AS. We don't want EVERYONE who registers to have access.

But then how do you know when a new user has registered?

Site admin always receives an email notification for new registrations, regardless of the registration type.

Is it possible to have them directed to a specified page after they have registered (to let them know their info is being reviewed).

Yes. In DNN under Admin | User Account select User Settings. In there you can select redirect after Registration (and redirect after login).