First, sharing the Joomla users and single sign-in are totally different things.
Second, to cross-check the user sharing, use the JMS tool menu and select the "table" tabs of the slave site that share another website.
You should find links between the MySQL users tables.
The list of tables that must be linked are the one that you can find in the JMS template / sharing tabs.
This depends on the joomla version.
You need at least the #__sessions tables, #__users tables and also the tables that are responsible of the ACL.
Review the tutorial video 16 that show a case on how to share the MySQL users.
www.jms2win.com/en/tutorial#jms12x-16
When the users are correctly shared, you should be able to modify the user name in one website and it would be reflected on the other website (as this is the same and unique data).