Transcription factors (TFs) interact with a multitude of binding sites on DNA and partner proteins inside cells. We investigate how nonspecific binding/unbinding to such decoy binding sites affects the magnitude and time-scale of random fluctuations in TF copy numbers arising from stochastic gene expression. A stochastic model of TF gene expression, together with decoy site interactions is formulated. Distributions for the total (bound and unbound) and free (unbound) TF levels are derived by analytically solving the chemical master equation under physiologically relevant assumptions. Our results show that increasing the number of decoy binding sides considerably reduces stochasticity in free TF copy numbers. The TF autocorrelation function reveals that decoy sites can either enhance or shorten the time-scale of TF fluctuations depending on model parameters. To understand how noise in TF abundances propagates downstream, a TF target gene is included in the model. Intriguingly, we find that noise in the expression of the target gene decreases with increasing decoy sites for linear TF-target protein dose-responses, even in regimes where decoy sites enhance TF autocorrelation times. Moreover, counterintuitive noise transmissions arise for nonlinear dose-responses. In summary, our study highlights the critical role of molecular sequestration by decoy binding sites in regulating the stochastic dynamics of TFs and target proteins at the single-cell level.