that the printer is connected to the desktop, then yes, the desktop needs to be on to print from the laptop.
It is possible to get kit such that the printerand both commputers are connected to a router. In this instance both computers do not need to be on. However, this is beyond my experience/expertise.
Given that the laptop is portable, might it not be a better solution to have a long enough lead to connect the printer direct to th elaptop?