Eukaryotic protein synthesis is an inherently stochastic process. This stochasticity stems not only from variations in cell content between cells but also from thermodynamic fluctuations in a single cell. Ultimately, these inherently stochastic processes manifest as noise in gene expression, where even genetically identical cells in the same environment exhibit variation in their protein abundances. In order to elucidate the underlying sources that contribute to gene expression noise, we quantify the contribution of each step within the process of protein synthesis along the central dogma. We uncouple gene expression at the transcriptional, translational, and post-translational level using custom engineered circuits stably integrated in human cells using CRISPR. We provide a generalized framework to approximate intrinsic and extrinsic noise in a population of cells expressing an unbalanced two-reporter system. Our decomposition shows that the majority of intrinsic fluctuations stem from transcription and that coupling the two genes along the central dogma forces the fluctuations to propagate and accumulate along the same path, resulting in increased observed global correlation between the products.