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.
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.
[1] https://taxsummaries.pwc.com/poland/individual/other-taxes