I am not sure if was understood what I want to point.

OK, when a webpage display the CB Login Module on all pages (sidebar), there is a another thing.
On my webpage is only a menu-entry to the CB Login Module.
But.
When i make for non logged in users only a menu-entry to "Community Builder » userprofile" User Profile (mandatory!), than link this for non logged in users to a error page. URL = /cb-profile .
understand?
I must also make a menu-entry for non logged in users to a CB Login module page. OK. But, when user is right logged in, the module redirect to the start page. URL = / . (First Hint: also i must set in the CB Login Module the "Login Redirection URL" to the "Community Builder » userprofile" User Profile (mandatory!), /cb-profile. Second Hint: when the page is not on /, than must set it on this place the Joomla base URL, example: /mysite/cb-profile.)
Also.
First: I must make two CB menu-entrys. One for CB Login. One for Profile view.
Second: I have also 2 Error Pages included.
- Menu: "Community Builder » userprofile" User Profile (mandatory!). "You are not logged in."-error page.
- Menu: "CB Login Module". "You are logged in."-error page.

The better way:
When user is non logged in, than:
Link the menu-entry "Community Builder » userprofile" User Profile (mandatory!) to the CB Login Module. Also not to the "You are not logged in."-error page.
When user is logged in, than:
Link the menu-entry "Community Builder » userprofile" User Profile (mandatory!) to the users profile.
Yet, i have only one link in my menu, the "Community Builder » userprofile" User Profile (mandatory!), and this have include the CB Login for non logged in users.

OK, there is really a little difficult.

Problem 1.)
Specially:
When i set a "Community Builder » userprofile (mandatory!)" under-menu-entry in the parent Community Builder » userprofile (mandatory!)" and name it "My Profile", than i have this Links/URLs:

"Community Builder » userprofile (mandatory!)" parent:
example.com/cb-profile/

"Community Builder » userprofile (mandatory!)" under-menu "My Profile":
example.com/cb-profile/cb-profile/

Other user profil pages:
example.com/cb-profile/cb-profile/userprofile/username

Solution for a "My Profile" menu-entry:
Don't set a under-menu "Community Builder » userprofile (mandatory!)".
Set a Menu Item Alias to the parent "Community Builder » userprofile (mandatory!)" menu-entry.

Problem 2.)
That's realy crazy with the menu-entrys!
When i set:
Community Builder » login / parent / Guest
Community Builder » userprofile (mandatory!) / parent / Registered
Than works the logged out redirect to click Community Builder » userprofile (mandatory!). And you don't see this menu-entry, wenn non logged in! (for testing i put the link URL in the browser).

example.com/cb-profile
redirect to --->
example.com/component/comprofiler/login

But, when i set the menu-entry:
Community Builder » userprofile (mandatory!) / parent / Public
and click this menu-entry as a non logged in user, than don't work the redirect to the CB Login. The link redirect to a Error Page.


On your test site:

The redirect to the login page works right for non logged in users.
When click the "Community Builder » userprofile (mandatory!)" parent menu-entry, than redirect to the CB Login Page (module).

The users profile page and other user profile page when logged in:
demo.cbdemosites.com/community-builder/profile

The users profile page and other user profile page redirect when non logged in:
demo.cbdemosites.com/component/users/?view=login


My 2 problems:
1) The Login Page for logged in users display an error "Error - You are not authorised to view this resource."
the same on your test site:
demo.cbdemosites.com/community-builder/login
(open the link after you logged in)
OK, you hidden this link, when user is logged in. (Guest Group?)

2) The "Community Builder » userprofile (mandatory!)" parent meny entry redirect not to the CB Login Page, but to a Error Page, when is set "Public".
example.com/cb-profile/
---> You don't have Access.

Also, all works fine only with a Joomla "Guest Group". There is no standard.

I wish for the "Community Builder » userprofile (mandatory!)":
When user is non logged in: redirect to the CB Login.
When user is logged in: redirect to CB Startpage (whatever you set).