By running its own network, based on Signal's protocol. Same thing goes with "private conversations" in Facebook's Messenger, or whatever it's called.
It's all the same protocol, but three different networks running that protocol. You can't communicate between these networks, and you'll need three accounts in order to communicate to people using these networks.
In fact, Whisper Systems has actively killed every third party app that tried plugging into Signal's network. While there's nothing stopping you from making a client, expect a lot of hostility from them if you decide to do so.
