I am a US citizen wishing to study for my master's degree in the Czech Republic. I have BS in computer science, so I would almost definitely qualify as a skilled worked for a Blue Card if that would be helpful. I can work as either a traditional employee or a 1099.
There are three separate questions here:
Are non-EU citizen students allowed to work in the Czech Republic? The answer is yes, you may work for as many hours as you want. See this page for proof:
No employment permit, Employee Card or Blue Card is required from a foreigner:
who is systematically preparing for a future occupation in the territory of the Czech Republic
Are non-EU citizens allowed to work for foreign companies? Absolutely, as long as they register as a self-employed person and pay taxes locally.
Can you continue paying taxes back in the US? No, as a full-time student you will become a tax resident of Czech Republic and here is where most of the tax would be paid. I am aware that the US taxes its citizens worldwide, but luckily for you there exists a tax treaty between the US and Czech Republic, allowing you to deduct Czech taxes from whatever you owe to the US government.