Both the server code and client code are open source. You're absolutely free to take them, modifly them, and do whatever you want with them. You're free to make your own fork and create your own chat service.

However, signal won't (and had not obligation to) provide your forks infrastructure. It costs money and time to maintain servers and they don't want 3rd party projects leeching their resources.

Again, your free to take their open source code and modify it and stand up your own infrastructure. If you do, you'll end up with your own chat service.