We present a detailed analysis of the interaction between two fluxon chains in parallel magnetically coupled long Josephson junctions, one of which is biased (''generator'') while another is unbiased (''detector''). The main effect is that the driven fluxon chain in the generator may drag the chain in the detector. We note that five different regimes of the interaction are possible: both chains may be pinned by the external magnetic field; both may move in a locked state, inducing the same de voltage in both junctions; in an unlocked state they may move at different velocities; the chain in the detector may remain pinned while the one in the generator is moving: and, finally, in a limited range of parameters the mean detector voltage may be negative, which implies that the detector chain is moving in the direction opposite to that of the chain in the generator. We consider a simplified model based on the assumptions that the fluxon chains are dense and rigid, and that their motion is nonrelativistic. In this model, each chain is represented by a single degree of freedom (its coordinate). Numerical and analytical consideration of the simplified model demonstrates that it is able to reproduce correctly all the dynamical regimes except for the negative-voltage one. To explain the existence of the latter regime, we introduce another model, suggested by the simulations, which is based on the presence of two fluxons and one antifluxon in the generator, and a single fluxon in the detector. The negative voltage is produced by motion of the antifluxon in a bound state with the detector's fluxon. The existence region of this state is limited by its collisions with free fluxons in the generator.