Hi, there! I am no expert on Shire money, but I know a bit about medieval currency. However, this is not the last word on things, but my personal opinion....
The Shire would likely have had copper pennies and haypennies, silver pennies (undoubtedly worth more) and something equivalent to a gold sovereign. Also possibly a three penney piece, four penney piece, and a six pence. The penney would have been the basic unit of coinage.
Tolkien mentions that "silver pennies" were paid out in the purchase of Bill the Pony. A pony is a fairly large purchase. Likely copper pennies would be used for food and for a night's lodging. Silver pennies would only be involved with larger purchases or a longer stay at the Inn. Gold coins would normally be reserved for larger transactions. You'd get a lot of change back if you paid for a night's lodgings with a gold coin!
The actual value of medieval coins was their weight in precious metal. This is unlike modern money, which is a "token" currency. However, Tolkien seems to have set the Shire in a "later period" than a place like Rohan so the money may have been more standardized. Of course, the guests at the Inn would also bring currency from their homelands so you probably had a real mishmash of different coins!
I'll try and check and see if I can see any links with more information. If anyone has more precise info, please go ahead and share it too. Hope this helps.
EDIT. Can't seem to find any pertinent links. The Shire Post website does sell "imaginery" shire coins of the type described above. Pio may possibly have more info on this (or she may not
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Cami Goodchild, Shire Mod