At first, this question sounded basic to me, but I couldn't find easy answer in Wikipedia. Since this SE prefers restricted scope, let's limit it to the empires of the 9th-10th century, e.g. the Carolingian Empire, the Byzantine Empire, the Abbasid Empire, or the Tang dynasty in China.
My assumption is that the government issued coins (exclusively?), and the coins would be used by its subjects for trade as well as to store wealth. How did the coins that were issued by the central government get into everyone's hands? One obvious way is through government expenditure, e.g. by paying people or buying things, someone other than the government would have coins, who would then pay someone else, and so on.
Was that all? My guess is that government expenditure would be a small part of the total trade, would the coins circulated this way be enough for everyone's trading and wealth-storing needs? Also, how about the corners of the empire, which might not have interacted much with the central government? Since the government collected taxes as well, wouldn't this also cause the money to be taken out of circulation?
Related Wikipedia articles (but none seemed to directly answer this question): Mint, Coin, History of Money.
(exclusively?)
, in many countries if you had bullion (gold, silver, etc.) you could go to a mint and get it converted into coins for a fee (seigniorage).