No, JChatSocial is not an app that manages Joomla login, so don't search a login after installing it.
The Joomla login is well managed by standard Joomla features, the login module and the users component, so there is no need to reinvent the wheel.
Moreover JChatSocial can work also for users that are not logged in. However keep in mind that to use chat with logged in registered users, it's needed a login to Joomla frontend. Being logged in to Joomla backend won't have the effect to be online on chat.
This happens because Joomla manages frontend and backend sessions in a separated way, and JChatSocial is a frontend app.