This paper introduces a multi-round interaction problem with privacy constraints between two agents that observe correlated data. The agents alternately share data with one another for a total of K rounds such that each agent initiates sharing over K/2 rounds. The interactions are modeled as a collection of K random mechanisms (mappings), one for each round. The goal is to jointly design the K private mechanisms to determine the set of all achievable distortion-leakage pairs at each agent. Arguing that a mutual information-based leakage metric can be appropriate for streaming data settings, this paper: (i) determines the set of all achievable distortion- leakage tuples ; (ii) shows that the K mechanisms allow for precisely composing the total privacy budget over K rounds without loss; and (ii) develops conditions under which interaction reduces the net leakage at both agents and illustrates it for a specific class of sources. The paper then focuses on log-loss distortion to better understand the effect on leakage of using a commonly used utility metric in learning theory. The resulting interaction problem leads to a non-convex sum-leakage-distortion optimization problem that can be viewed as an interactive version of the information bottleneck problem. A new merge-and-search algorithm that extends the classical agglomerative information bottleneck algorithm to the interactive setting is introduced to determine a provable locally optimal solution. Finally, the benefit of interaction under log-loss is illustrated for specific source classes and the optimality of one-shot is proved for Gaussian sources under both mean-square and log-loss distortions constraints.