Payroll Companies are globally registered companies (or companies registered in a specific country), that allow employers to fully employ people in that specific country. While it's still common to hire remote employees as contractors, in this case, the employee is lawfully employed in their home country, including all benefits that come with that, and all payroll and invoicing goes through the payroll company.
In some cases, that payroll company will also be responsible for reimbursements, management of PTO days, benefits etc.
Payroll companies often only forward invoice payments, and take a $100-$300 cut per month.
Usually it works out to bring on a remote employee as a contractor. The "employee" is registered in their home country as self-employed, and issues an invoice to the employer every month. Benefits are invoiced too, or provided directly, so are reimbursements, bonus payments, ...
In some cases, this either doesn't work or is not favourable:
In the following interactive sheet, we have a continuously updated list of countries and their respective payroll companies (and labour laws).
Hiring employees in your own country is fairly simple. The only real differences may be between state or regional taxes and regulations. But if you want to hire remote workers in foreign countries as employees or contractors, then it becomes more complex.
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