The study of almost surely discrete random probability measures is an active line of research in Bayesian nonparametrics. The idea of assuming interaction among the atoms of a random probability measure has recently spurred significant interest in the context of Bayesian mixture models, allowing the definition of priors that encourage well-separated and interpretable clusters. In this talk, we provide a unified framework for the construction and Bayesian analysis of random probability measures with interacting atoms, encompassing both repulsive and attractive behaviors. We develop a full Bayesian analysis without making any assumptions about the finite point process that governs the atoms of the random measure. We then focus on a clever choice of the underlying finite point process, leading to shot-noise Cox process mixture models. We show that assuming a shot-noise Cox process for the mixture locations yields tractable theory, efficient algorithms, and a novel notion of clusters that may consist of multiple mixture components with similar parameters. We also demonstrate how this construction can be extended to cluster observations divided into groups, giving rise to the hierarchical shot-noise Cox process (HSNCP) mixture model. Previously proposed models allow for clustering across groups by sharing atoms in the group-specific mixing measures. However, exact atom sharing can be overly rigid when groups differ subtly, introducing a trade-off between clustering and density estimation and fragmenting across-group clusters. We show how the HSNCP overcomes this density-clustering trade-off. Simulation studies and a real data application showcase the usefulness of our proposal.