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Usually, no, and yes.
Some coins (Greece, for example) were outsourced early on when the large quantities were being minted for the eruro’s introduction. They have littler mintmarks in one star to show if they were one of these. Some other countries also outsourced some of their minting, without any special designation on the coin (but I know some 2 euro coins have the wring edge inscriptions as a result of the contracted country using it’s own lettering instead of that for the customer country.)
They are otherwise minted by each country (except maybe for the smallest countries, which might outsource).
Some contain “privy marks” showing the mintmaster or other person, but I personally do not consider these mintmarks as such, as all the coins have the same mark.,
In the case of Germany, however, there are five mints, and each has it’s own mintmark. As far as I know, no other country has multiple mints, or at least they do not have individual mintmarks.
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