Project Matrix is an extremely interesting concept about open and federated networking with strong emphasis on collaboration and privacy. You can compare the capabilities of the Matrix servers and clients to the ones of commercial collaboration applications that offer instant messaging - incl. concepts such as spaces, channels/rooms, threads, etc., voice and video calls.
And yes, you can own your own Matrix server and use it to keep you conversation safe and private. There is a wide number of open source client application for all major desktop and mobile platforms.
The Liberated Edge recommends everyone using Dendrite as the Matrix server solution of choice for the Edge.
Evaluate your needs and make up your mind how you will be using the Matrix. Dendrite will facilitate many of the collaboration workflows that you may already know from commercial apps.
Dendrite is running as a standalone API server exposing Matrix API endpoints.
All of the clients besides SchildiChat offer desktop applications for Linux, macOS and Microsoft Windows.
We have evaluated three server solutions implementing the Matrix specification to varying degree. We have picked up to support Dendrite as a Matrix server as it is lightweight and can run well even on less powerful hardware, e.g. Raspberry Pi. At the same time the project tracks their compatibility with the Matrix specification and works on improving it.
There are issues with cross signing on our test systems that run Dendrite. Keep in mind that if you'd sign out from your session in Element, you may not be able to reconstruct your chat history.
These are the evaluated solutions.