Structured data in the form of networks are increasingly common in a number of fields, including the social sciences, biology, physics, computer science, and many others. A key task in network analysis is community detection, which typically consists of dividing the nodes into groups such that nodes within a group are strongly connected, while connections between groups are relatively scarce. A generative model well suited for the formation of such communities is the assortative stochastic block model (SBM), which prescribes a higher probability of a connection between nodes belonging to the same block rather than to different blocks. A recent line of work has utilized Bayesian nonparametric methods to recover communities in the SBM by placing a prior distribution on the number of blocks and estimating block assignments via collapsed Gibbs samplers. However, efficiently incorporating the assortativity constraint through the prior remains an open problem. In this work, we address this gap by studying the effect of enforcing assortativity on Bayesian community detection and identifying the scenarios in which it pays dividends in comparison with standard SBM. We illustrate our findings through an extensive simulation study.