At least, you would pay Portuguese taxes for that employee and you would likely need to register with Portugal as an employer.
You'd be in a strange split-brain situation, requiring legal advice. I imagine you could file income taxes in Denmark, and these would be deducted from the worker's Portuguese due taxes under the double taxation prevention agreements.
However, the employee will have to pay taxes where he lives, independent for whom he works for.
Not a lawyer, but I think that's it, outside of paying the appropriate employment taxes and fees to the employee's state of residence/operation.
I'm guessing this is what the article's author is referring to, but the way he explains it sounds like someone advised him to incorporate wholly owned subsidies in the states where he had employees, which would be both unnecessary and more expensive.
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.