What exactly is "the rotating lighthouse model" in the context of a double pulsar?
It seems that the authors are just referring to the accepted model of a pulsar, i.e. a neutron star spinning and emitting beams of radiation at its poles. In that sense, the term is used here no differently than it would be in the context of an isolated pulsar. It's just a very simple way of visualizing why an observer far away appears to see periodic pulses rather than constant emission.
If the lighthouse model is correct, then the beams from one pulsar may interact with the other pulsar, which in turn would cause changes in the second pulsar's emission each time the beam sweeps across it. The modulation period would then be approximately equal to the rotational period of the second pulsar - not precisely, though, because of orbital motion; hence, the comparison between Earth's sidereal day and solar dat.
Modulation of pulse strength and features therefore isn't a feature of the lighthouse model but rather a consequence of it, present in some binary pulsar systems but of course not in isolated pulsar emission.