miRNAs are small RNAs that regulate gene expression post-transcriptionally. By repressing the translation and promoting the degradation of target mRNAs, miRNAs may reduce the cell-to-cell variability in protein expression, induce correlations between target expression levels, and provide a layer through which targets can influence each other’s expression as “competing RNAs” (ceRNAs). However, experimental evidence for these behaviors is limited. Combining mathematical modeling with RNA sequencing of individual human embryonic kidney cells in which the expression of two distinct miRNAs was induced over a wide range, we have inferred parameters describing the response of hundreds of miRNA targets to miRNA induction. Individual targets have widely different response dynamics, and only a small proportion of predicted targets exhibit high sensitivity to miRNA induction. Our data reveal for the first time the response parameters of the entire network of endogenous miRNA targets to miRNA induction, demonstrating that miRNAs correlate target expression and at the same time increase the variability in expression of individual targets across cells. The approach is generalizable to other miRNAs and post-transcriptional regulators to improve the understanding of gene expression dynamics in individual cell types.