We know in Bilgewater that the people use Silver Serpents as a currency from GP adding a few to his bounty on the bounty board. That may be only one denomination of cash named in all of League of Legends lore... "Golden Krakens" comes to mind, I believe I read that in one of the stories maybe, could just be me taking Silver Serpents and making the next logical conclusion though.
Point is, for the most part the economies of the other regions isn't touched on at all though.
An insular society like Demacia ought to have a state regulatated centralized currency that out-values any other money brought into the region. With any of the corresponding Jarvans' face on one side of their coins and either Demacian Wings or The Sword of The Protector on the other. Maybe they could actually be minted from petricite thats mixed in with any number of metals. Which would probably make any transaction for Lux very uncomfortable.
A multicultural society like Noxus probably accepts all forms of currency from every inch of their empire, Shurima and Valorans' minor kingdoms like Arbormark. I'd imagine they have conversion rates for all of these currency based on the Noxian mint and every merchant must be prepared to count coins from a hundred different corners of the world to close out any sale.
The Freljordians may just trade survival supplies, pelts, meats and weapons and forego any form of currency all together. Relying on their intricate tribal bonds and the need to rely on each other for survival to over any idea of recompense for services.
Ionia? Ixtal? The Buhru? Piltover and Zaun?
Just gets the mind going thinking about these things. Any Rioters able to weigh in?