As far as legal obligations go, that's mostly going to depend on the country you're in, and how they're engaging you. If you're a full-time employee with an unlimited contract, ideally they'd employ you under their local legal entity. If they don't have one, they might use an "Employer of Record" service like Remote.com (never used them myself, and there are other companies that offer similar services).

If, however, you're a "contractor" but with an unlimited contract (or a contracted renewed repeatedly), and potentially a few other things that would give the impression you're an employee, then you might have some issues - I know that Germany will make your life miserable if you're determined to be engaging in "Scheinselbstständigkeit (or arbeitnehmerähnliche Selbstständiger)," or "Bogus self-employment (or quasi-employee self-employed)." The way around that is to file taxes as if you were employed and pay all the taxes that the company would ordinarily pay, which means that you need to account for that in your negotiations, plus the costs of a tax accountant if you're not comfortable doing that yourself. I don't know which country you're in, but it's something to look into.

If you're a genuine contractor who regularly switches companies (or at least multiple regular clients, depending on local law/etc) then I don't think there are any issues that would be specific to the US - you'd have to deal with things like VAT, currency conversion issues, possibly different tax years, etc, etc, but I'd also expect those to be common issues with any local contractor who works with clients in other countries.