Hi peachocadmin and screenfree,
Private blog will always require users to sign up for a WordPress.com account. See this page:
All viewers added to a private blog must have a WordPress.com account—this ensures that only those you’ve authorized to see your blog are the ones viewing it:
Here are the instructions for inviting viewers:
peacockadmin, the checkmarks you are referring to in your post apply only to people commenting on your content, not to viewing a private blog.
screenfree, I don't think you were supposed to receive an email, only the user who was trying to register. Once they create their account, they should receive a confirmation email. Please let them know to check their spam folder in case the email got falsely caught in the spam filters. Once they click the confirmation link in that email, they will have access to your site. Please note that after confirming their account, the invitee will have to go back to the invitation e-mail and click on "Accept invitation" again!
They will need to click the “Accept Invitation” button in the invitation email. If they are logged into their WordPress.com account, they will be taken directly to your blog. If not, they will be taken to the log-in screen, where they can either log in or create a new user account.
Note that users who create a new WordPress.com account at that point will still need to return to the invitation email and click the “Accept Invitation” button after doing so.
Hope that helps!