Earlier quoted context omitted.
Hiring an employee in another state may create a nexus in that state, and this may require you to file income tax in that state. For example, if you incorporate in Delaware, and then hire someone for >$50k in Texas, you may have to file company income taxes in Texas. Not that Texas income attributed to your company (or partners) will be much, but, computing and filing is best done by hiring an accountant.
I'm not sure about EU law - but if you have a company in Denmark, and want to hire someone who works in Portugal as a permanent employee, and not a contractor, what do you need to do?
However, the employee will have to pay taxes where he lives, independent for whom he works for.