I'm located in the suburbs of Frankfurt/Main. The pharmacist took a cursory glance at my national ID card (I have primary residency in Germany) to make sure it matched the passport, but that's it.

My hunch it that each country participating in the scheme is responsible and allows just its residents and citizens. Its main purpose wasn't directed for non-Schengen/particpating citizens or residents, although I'm sure down the road the CE might try and do reciprocity agreements. It could be more difficult for an outside tourist who got both doses outside the EU, because the pharmacist essentially has the burden of verifying the authenticity of the vaccination record, something risk-averse Germans don't willingly do.