As an EU citizen you have the right to reside in another EU country, provided that you are either working or you have comprehensive health insurance and sufficient resources not to be a burden on the host country's social security system. In principle, I don't believe you have to be working in your host country to qualify under the "worker" condition. However, national health insurance / social security schemes are not well joined up, so you may find it easiest to take out private health insurance anyway.
In my personal experience, when I (UK national) was living in Spain and employed by a UK company I didn't have national health coverage in either country: I didn't qualify to register with the UK's NHS because I wasn't resident in the UK, but I couldn't get a social security number in Spain without either a contract from a Spanish company or proof of being self-employed.
Taxes can be interesting in this situation. I was being paid in GBP, but I had to declare my income in EUR so that Spain could tax me. The fun part was the exchange rate: I had to find an official exchange rate for each payday and apply it to the earnings from that payslip. If you're being paid in USD then you'll need to get advice from a local accountant on how you need to calculate income for local tax purposes.