Check the script that is called and its return when you login on the website.
When you are logged, you should have a call with the login and have a return = true.
I suppose that you have a return false.
If you have a return false, verify your settings, also verify the secret value between the websites.
Do you use JMS with SSI Domain ?
Which Joomla version do you use ?
I suggest that you also review the SSI domain video that how to cross-check you configuration.
tutorial.jms2win.com/joomla/multisite-12x/ssidomains
At min 22