Is it possible? Have a second reg form with User that goes to a specific group as opposed to main site signup. Is it possible to specify on this reg form alone that an Admin gets a mail on new signups?
Unfortunately this isn't possible with User. I recommend submitting this as a Feature Request under the specific add-on category: http://support.solspace.com/support/discussions
The only option for notifying to a specific email address when a registration is completed is through EE's native settings under Member => Preferences => Notification Preferences: "Send new member notifications?" and "Email Address for Notification". However, this is of course not specific per-registration form.