Correlated neuronal activity is an important feature in many neural codes, a neural correlate of a variety of cognitive states, as well as a signature of several disease states in the nervous system. The cellular and circuit mechanics of neural correlations is a vibrant area of research. Synapses throughout the cortex exhibit a form of short-term depression where increased presynaptic firing rates deplete neurotransmitter vesicles, which transiently reduces synaptic efficacy. The release and recovery of these vesicles are inherently stochastic, and this stochasticity introduces variability into the conductance elicited by depressing synapses. The impact of spiking and subthreshold membrane dynamics on the transfer of neuronal correlations has been studied intensively, but an investigation of the impact of short-term synaptic depression and stochastic vesicle dynamics on correlation transfer is lacking. We find that short-term synaptic depression and stochastic vesicle dynamics can substantially reduce correlations, shape the timescale over which these correlations occur, and alter the dependence of spiking correlations on firing rate. Our results show that short-term depression and stochastic vesicle dynamics need to be taken into account when modeling correlations in neuronal populations.