A wireless printer is a small network device. Your computer finds it by address, sends the document over the network, and the printer stores and prints it. Nothing is sent through the internet, which is why printing works when the broadband is down but the network is up.
The weak point is the address. Many routers hand out addresses that change over time, so a printer set up on Monday can be unreachable by Friday even though nothing was touched. The computer still holds the old address and reports the printer as offline.
Discovery is the second weak point. Devices find printers by broadcasting on the local network, and those broadcasts do not cross between networks. A laptop on the guest network or on 5 GHz separated from the printer's 2.4 GHz network will simply never see it.
Finally, the queue can stall. One failed job at the front blocks everything behind it, so later documents pile up silently and nothing prints even after the original problem is fixed.