That is actually a little like how US money used to work--badly--before the civil war. A storekeeper would keep a third-party guide behind the counter, regularly published for their city or region. The guide would help them recognize the designs of private notes from different banks, and contain recommendations for what value (if any) someone could place on the note depending on the reputation of the bank and how far away they'd have to go to the nearest branch.
"It costs $3? Okay, here's a $5 from the Far South Bank."
"That's a long ways from here in Middleville, traveler, I'll take it as $4."
"If you can give me $2 from Extreme North Bank for change, then I'll take the deal. Otherwise $1 for something local isn't as useful for me since I'm leaving tomorrow."
"Sorry, today I can only make change in local stuff or Eastern Railroad."