Hypergraphs are widely adopted tools to examine systems with higher-order interactions. Despite recent advancements in methods for community detection in these systems, we still lack a theoretical analysis of their detectability limits. Here, we derive closed-form bounds for community detection in hypergraphs. Using a Message-Passing formulation, we demonstrate that detectability depends on hypergraphs' structural properties, such as the distribution of hyperedge sizes or their assortativity. Our formulation enables a characterization of the entropy of a hypergraph in relation to that of its clique expansion, showing that community detection is enhanced when hyperedges highly overlap on pairs of nodes. We develop an efficient Message-Passing algorithm to learn communities and model parameters on large systems. Additionally, we devise an exact sampling routine to generate synthetic data from our probabilistic model. With these methods, we numerically investigate the boundaries of community detection in synthetic datasets, and extract communities from real systems. Our results extend the understanding of the limits of community detection in hypergraphs and introduce flexible mathematical tools to study systems with higher-order interactions.