@mingu With printers its best not to use the manufacturer's garbage driver and use IPP instead (although CUPS may not easily allow you to set IPP or IPPS instead of a specific driver).

In the CUPS settings, if you select "share this printer" and the computer is setup to work with mDNS and zeroconf, the printer will be available over the network via the computer via IPPS - although that requires a server or the computer being on whenever you want to print something from an external computer.

I guess some printers are broken until you connect to the printers webserver and change the settings so that IPPS is enabled and "Wi-Fi direct" is disabled.

>I guess you need to make the printer to connect to the internet in the first place
Wireless LAN printing doesn't require an internet connection - if the printer does, then it's crucial that the printer is never connected to the internet no matter how much it asks (i.e. a VLAN without internet access).

To work over a LAN, the printer does need to be connected to the LAN, although a wired connection is recommended, as Wi-Fi doesn't perform well when you're trying to send out packets at the exact same time packages are going into another device - otherwise the only option is to connect to the printer's Wi-Fi network and use "Wi-Fi direct".