Coherent dual comb spectroscopy can provide high-resolution, high-accuracy measurements of a sample response in both magnitude and phase. We discuss the achievable signal-to-noise ratio (SNR) due to both additive white noise and multiplicative noise, and the corresponding sensitivity limit for trace gas detection. We show that sequential acquisition of the overall spectrum through a tunable filter, or parallel acquisition of the overall spectrum through a detector array, can significantly improve the SNR under some circumstances. We identify a useful figure of merit as the quality factor, equal to the product of the SNR, normalized by the square root of the acquisition time, and the number of resolved frequency elements. For a single detector and fiber-laser based system, this quality factor is 10(6) - 10(7) Hz(1/2).