When I am using "https://security.stackexchange.com" there appears to be no "https" link on two buttons: "Chat" and "Blog".
So if I am logged-in and I go to "Chat" it says I need to logon. However the "Log-In" button doesn't work I think because I am already logged-on the "https". By clicking it it just redirects me back to the page I was on.
See screenshot below:
stackexchange.com
domain are allowed for unsecured connections here, you may want to check this on your side too)..stackexchange.com
cookies are allowed for any kind of connection (both secure and unsecure). Have you checked is these cookies were defined the same way on your system? What you describe is the usual symptom for session cookies being restricted to secured connection only.