Your question and request for help is a good one. I've been there and done that. IMO, the easiest way to accomplish what you are trying to do is use Proscan software, where all of this is easy to set-up and done for you by the software. "However", seeing as how you don't have a Uniden radio, that won't work for you LOL.
You could use Win500 software right now for a one-to-one private stream. You run the stream using the Win500 server software running on a PC at home and you connect to the stream privately over the internet using the Win500 client software. You still need to configure your home network router to port forward the Win500 connection from the outside and/or configure the port being used as an exception in Windows firewall. Once this done you will have exactly what you want, but only for one listener at a time. If you are technically inclined, the author of Win500 software published a free utility called TCPMux, which can be used to get around the 1 connection limitation when using Win500. It is however a technical "adventure" IMO. There is a thread here that speaks to this configuration.
Win500 TCPMux Configuration Instructions
The other alternative (which I often use), is to run Icecast for Windows (freeware) as a server to provide the stream and then broadcast it out using ScannerCast or Radio Feed. The proper configuration of port forwarding on your router is still a requirement. The URL that you use for the stream becomes the private address that you and others use to connect & listen.
All of this can be done and many of us already do. In my experience, there is some trial & error and some basic knowledge of networks/routers/firewalls required along the way......