Can Safari receive desktop notifications from YouTube or Facebook?
If you go to your Messenger messages on Facebook, at the bottom, click on "show in Messenger". Then click on the gear icon and go to settings. You can enable notifications there, but I don't know if this will also enable Facebook messages, it will ask you if you want to receive messages from "facebook.com", so I guess it should work. 
Most websites ask you if they can send messages the first time you open that website, but for some reason Facebook and YouTube don't do that.