In America, cash production is only legally done by one centralized entity, and it is done by them on a regular basis. As far as I know, this holds true in all developed nations.

Banks are allowed to emit debt at the condition that a fraction of it is backed by some liquidity.

Banks decide where and when to create this debt so yes, money production is indeed decentralized.

If you’re referring to actual coinage and notes then yes, facilities are indeed centralized, but with electronic debit cards the lines are blurred further

However, let's say I run a dollar store. Can I afford to do an exhaustive verification for every dollar I receive? maybe not.

There are security features built into each bill that help me establishing if it's legit or not, that I can do myself without third parties.