From molecules in cells to organisms in ecosystems, biological populations fluctuate due to the intrinsic randomness of individual events and the extrinsic influence of changing environments. The combined effect is often too complex for effective analysis, and many studies therefore make simplifying assumptions, for example ignoring either intrinsic or extrinsic effects to reduce the number of model assumptions. Here we mathematically demonstrate how two identical and independent reporters embedded in a shared fluctuating environment can be used to identify intrinsic and extrinsic noise terms, but also how these contributions are qualitatively and quantitatively different from what has been previously reported. Furthermore, we show for which classes of biological systems the noise contributions identified by dual-reporter methods correspond to the noise contributions predicted by correct stochastic models of either intrinsic or extrinsic mechanisms. We find that for broad classes of systems, the extrinsic noise from the dual-reporter method can be rigorously analyzed using models that ignore intrinsic stochasticity. In contrast, the intrinsic noise can be rigorously analyzed using models that ignore extrinsic stochasticity only under very special conditions that rarely hold in biology. Testing whether the conditions are met is rarely possible and the dual-reporter method may thus produce flawed conclusions about the properties of the system, particularly about the intrinsic noise. Our results contribute toward establishing a rigorous framework to analyze dynamically fluctuating biological systems.