Yes, the phone has to be connected to the internet for it to work. It says so in the documentation, and on the very first page "To use the Signal desktop app, Signal must first be installed on your phone."
Suppose I install and configure Signal/Phone, then install and configure Signal/Desktop, then turn off the phone.
Will Signal/Desktop still work while the phone is turned off?
* If yes, then what you say appears to be wrong.
* If no, then that's genuinely astonishing to me, and I've learning something.
Is there a grace period ?