We investigate the dynamics of a bimolecular reaction on conically intersecting potential energy surfaces. The flux operator method of calculating the state-specific total reaction probability is extended to a coupled-surface problem, both in the diabatic and adiabatic electronic representations. The reaction probabilities are calculated from their expectation values with the aid of a time-dependent wave packet (WP) approach. The initial WP is prepared in an adiabatic electronic state, and it is propagated in a suitable diabatic electronic representation The initial state-specific and energy-resolved reaction probability is given in analytical forms in both the adiabatic and diabatic picture. The diagonal correction (Born-Huang term) to the uncoupled adiabatic (Born-Oppenheimer) Hamiltonian is discussed. The above formalism is applied to the H + H-2 exchange reaction on its conically intersecting double many-body expansion (DMBE) potential energy surfaces. We report the initial state-selected reaction probabilities for energies extending up to the onset of the three-body dissociation of this system, We find only a minor impact of the conical intersection on the reactive scattering dynamics of H + H-2. A closer inspection of the electronic population reveals a very small fraction of the WP traversing the upper adiabatic sheet during the course of the reaction. The accuracy of the DMBE potential energy surface is assessed by comparing with new ab initio data.