Any American expats here working as 1099 independent employee?

The US company must issue you a 1099 if you are working as an individual. And you must show to the US company that your payments are not subject to backup withholdings since you are living abroad.  You may need to submit a W-8 form.

For what it is worth, if you live full time in Hungary more than 180 days, you must declare Hungarian residency. And if you live and work in Hungary you are subject to Hungarian tax law. For sole proprietors that normally means forming or register a Hungarian business to work as a sole proprietor (which requires an accountant by the way -- even for a sole proprietor unless you declare KATA status).

That is, you can not really live and work in Hungary for 2 years as simply an "American" sole proprietor. Sure, some may try it, and get away with it, but not "kosher". If you live in Hungary, your tax nexus as well as any work income is in Hungary. Which means, at a minimum, a 35% and as much as a 50% tax on your income here as a sole proprietor (which is why most people form a BT company, a KFT company or use KATA).

There are some details in the US-Hungarian tax treaty that allow exemptions for some type of income, but if you are a dual citizen, and are a sole proprietor, you are probably not entitled to them. Hungary can tax it citizens fully on world wide income if they live in Hungary.