That's sadly not the case: EU citizens can work everywhere in Europe, but employing people across Euro countries is not easy at all. You need legal advice to comply with both your country's and the employee's country's laws, some countries require an actual registered office/entity, the employee protections e.g. in Italy and France make it very hard too.

This is for employment, of course. You can contract someone from anywhere, regardless of EU or not.

Seems like I've been wrong then, sorry. This was the impression I got from our HR office (it is true that we require relocation).