Well, from someone else running Cinnamon on Arch Linux, browser notifications from the latest version of Firefox do work. Never had a problem with them, which I realise is unhelpful.
I was running Mint before I moved to Arch Linux, and I just moved my home partition over to the new system. Perhaps Mint writes some configuration file that Arch reads but can't create itself? It would be a home-directory file, not a system one, in that case.