I don't want to sound negative but with current tax legislation in most EU countries, this is far more complicated than it may sound, at least to do it lawfully. Not impossible but in many cases, it requires a legal entity in your home country to be fully employed (i.e. not freelancer-based).
I investigated a somewhat related scenario: Being employed at a German legal entity with an already-remote contract and relocating to a neighboring EU country for permanent remote work. Turns out to be complicated and quite unattractive for your employer to support this, due to national tax+labor laws, potentially mandatory social insurance, etc.
It's not that complicated actually. There are already companies who establish local companies that function as intermediary for hiring local people. Alternatively, you can setup your own company that will employ you, pay the local taxes etc. and the company that actually you work for will pay your company for purchasing services.
Deel, Oyster HR and Papaya Global are examples for the first solution.
Though VAT is one of the easier things in practice once you have everything running. Granted I used an accountant to deal with all the taxes, which totally is worth the cost.
As stated, I am only focused on employment, not freelancing. See also parallel discussion with challenges of contractor vs employee and false self-employment.
Disclaimer: IANAL and I am not a tax accountant. This is not tax advice. See a local professional.
Direct employment by a U.S. company is possible and legal in Europe/EU. The company doesn't need to establish a local branch, but needs to get a tax number (so that the tax office can track the tax payments you make every month, because instead of automatic withholding of taxes typically done by a local employer, you will be responsible to do them yourself). Talk to a local tax accountant, they can typically set your U.S. employer up with local social security and the tax office. You will be taxed according to your local tax rates, same for social security contributions. Often that includes "employer contributions", which are charged on top of your "gross" salary. So the cost to your employer is your gross salary plus employer contributions to social security, which includes pension insurance, disability insurance, accident insurance, labor fund, and sickness insurance. According to [1], this can range between 19.21% to 22.41% on top of your gross salary.
Generally, setting up a tax number and getting enrolled in the social security system is a pain and (depending on the country and local government office employees) may require documents signed by high-level people of the company. Therefore, not a lot of U.S. companies are likely willing to go down that road for a single employee.
You are out of luck then, there is no legal framework for employment cross-border anywhere. Maybe in Dubai with their remote work visa giving you 0% taxes but that would require a relocation there.
Not true at all! My (EU) tax form has a special field just for this case - income from out-of-EU employment... (and because of tax residency rules that applies only if I live there)
And there definitely is a legal framework - this has very exact rules specified by the bilateral tax/trade agreements, every combination of western countries has them, and many non-western countries too.
I investigated a somewhat related scenario: Being employed at a German legal entity with an already-remote contract and relocating to a neighboring EU country for permanent remote work. Turns out to be complicated and quite unattractive for your employer to support this, due to national tax+labor laws, potentially mandatory social insurance, etc.
Would appreciate any hints though.