Opposite: You need a tax id for the country the employee is in because you pay the employer part of their local health insurance, pension and whatever other benefit the employee is entitled to in their country. The contract needs to be bi-lingual (certified). All local laws like vacation days, sick days, maternaty leave rules, which public holidays are observed need to be respected.
Most companies don't want this extra hassle and instead make the remote person a contractor, basically receive a single monthly bill. The contractor is responsible to file their income tax, get health insurance etc.
An alternative is for the remote person to get employed by a local agency, who then sends a monthly bill to the US company.
Another option is to open a subsidary. Of course hardly worth it for one employee.
In Europe if a remote employee (let's say Greece) travels to their employer (let's say in France) to work or have meetings they have to pay taxes in France for those days. Not an issue if it's less than 5% of the working days per year. If it's more than 10% the employee has to file taxes in both countries. I've been in this situation. This is Europe, not the US, but shows how crazy complex tax situations can become.
