Do pulsar beams interact with the interstellar medium? Pulsars are rotating objects that emit a beam in a direction that is not aligned with the axis of rotation. Because of this, we see them as a periodic pulse. If the beam isn't towards us, we don't see it. Since we need to be on the path of the beam to see it and the beam is only a few degrees wide, it is believed that there are many objects that are pulsars, but that we see as ordinary neutron stars (or can't see at all). 

Does the beam from a neutron star interact with the surrounding medium in such a way that we might see the result of this interaction? The same way you can't see a laser beam, but in a very dusty room, you can see the light from the laser bouncing off of the dust particles, making it possible to indirectly see the beam.
Yes. the emissions from a pulsar do interact with interstellar medium - this effect is called dispersion. When the emission from the pulsar hits free electrons (and dust) in the interstellar medium, different frequencies in the pulse become "delayed" by different amounts corresponding to their frequencies - specifically, lower frequencies are delayed more than higher frequencies. So, when we receive the signal from a pulsar in a radio telescope, the pulse is "dispersed", or curved. Software is used to de-disperse these pulses by testing out many different "dispersion measure" values to line up the emissions correctly at different frequencies. You can find that here. 

This is the main way we can be sure the pulses interact with the interstellar medium. Based on how dispersed those pulses are, and how much "stuff" we think the interstellar medium contains, we can get a very basic idea of how far away the pulsar is when we accurately calculate its dispersion measure.

I don't, however, know of any effects like what you referred to with the laser, resulting in an ability to "indirectly" observe pulsars... the pulses themselves are already quite weak and "seeing" a pulsar directly takes a decent amount of time on a telescope along with large amounts of data processing. From my personal experience in the field along with taking many observations on radio telescopes designed to observe pulsars, I highly doubt that the laser-pointer effect you described is observable.

For more details on pulsar dispersion, take a look here.