In the last decade, subgraph detection and enumeration have emerged as central problems in distributed graph algorithms. This is largely due to the problems' theoretical challenges and practical applications. In this paper, we initiate the systematic study of distributed sub-hypergraph enumeration in hypergraphs. To this end, we (1) introduce several computational models for hypergraphs that generalize the CONGEST model for graphs and evaluate their relative computational power, (2) devise algorithms for distributed triangle and simplex enumeration in our computational models and prove their optimality in two such models by showing matching lower bounds, (3) introduce classes of sparse and "everywhere sparse" hypergraphs and describe efficient distributed algorithms for triangle and simplex enumeration in these classes, and (4) describe general techniques that we believe to be useful for designing efficient algorithms in our hypergraph models.