I don't, because there's a bilateral US-Denmark "totalization agreement" on social security, which ensures no double-taxation. There's a rubric determining which country you owe taxes to, depending on where you reside, what your nationality is, and who your employer is (but it's never "both"). It also takes a small step towards allowing you to shift years accumulated in one country's program to the other.
Such treaties exist with 24 countries, so of course you'd be right if I didn't live/work in one of those 24: http://www.ssa.gov/international/agreements_overview.html