It links to your phone to authenticate you. But from there, all messages are sent and received directly from the servers to all clients. So if your phone is off, but your desktop is on, the desktop client still receives them.
The catch is that if your desktop is off, it won't be able to "catch up" later on any messages that it missed. Although I don't see why that's impossible to implement in principle.
Regarding Signal, they say this on their website:
> Signal Desktop is a Chrome app which links with your phone, so all incoming and outgoing messages are displayed consistently on all your devices.
https://signal.org/blog/signal-desktop/
That seems similar to what WhatsApp does, but I haven’t tried myself. Do you have more details on how that would work without passing by the mobile app?