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.

About WEBRTC and HTML5 peer-to-peer technology you could be aware of STUN and TURN servers involved in the process. This doesn't mean that data are streamlined on third party servers. STUN and TURN servers are public web services used in the WEBRTC architecture only to find IP addresses and start the direct communication between 2 peers when they are behind routers and firewalls. More info can be found on Mozilla network: https://developer.mozilla.org/en-US/docs/Web/Guide/API/WebRTC/WebRTC_architecture