Yes, JChatSocial is a fully non-hosted chat solution for Joomla, because it doesn't require intermediate servers and resources but only the plain Joomla installation.
All the messages and data exchange between users are hold directly in the Joomla database. You can access and export data using Joomla backend if needed.
The videochat available in the Enterprise edition, is based on the innovative peer-to-peer technology that allows direct data stream between 2 browsers. This means that all video and audio data are not delivered using intermediate servers and no software installations on third party servers are involved.