The band structure of two coupled one-dimensional (1D) bicomponent magnonic crystals is constructed from theory. The model takes into account both the exchange interaction and the long-range dipolar interaction between the magnonic crystals, for which different coupling configurations are analyzed systematically. In comparison with a monolayer 1D bicomponent magnonic crystal, the band structure of the bilayer magnonic crystal is significantly modified and depends on the geometric positioning between the layers. The widths of the band gaps may be altered considerably to induce a flattening of the high-frequency modes or nonreciprocity in frequency of two counterpropagating collective spin waves. In the latter case, the nonreciprocity is mainly observed in modes with frequencies above the first band gap, whereas the low-frequency modes remain almost reciprocal. These theoretical results show that two coupled magnonic crystals are excellent candidates to assist in controlling the collective spin waves, thereby permitting great versatility in designing reconfigurable spin-based devices.