Adding a user creates the account straight away, with no email or invitation. To have people register themselves instead, see Invite People to Intranet.
After you click Register, you return to Manage People, where the new user appears at the top of the user list. To add more information to the user's profile, click the gear icon > Edit (see Edit User).
If a username or email address is already taken: When you move out of the username or email field, the form marks it Already taken, choose another if another account already uses it, and a deactivated account counts. Look for the old account under Deactivated People, then either reactivate it instead of adding a new one, or delete it (see Delete User) and add the person again.
To set the default privacy level for new user profiles, go to Control Panel > System > System Properties and set the NewUserProfilePrivacy property. This system property (a site-wide setting) decides whether a new profile starts out public or private:
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