Will line conditioner do anything for the other receptacle in a duplex?

It should, as the these pieces are designed to take a "dirty" incoming signal, and send out a "clean" one.  The "clean" signal it sends out goes to your gear *AND* back to the outlet/duplex, hence a "clean" duplex (theoretically a whole circuit).  The effect will *not* be as strong on the duplex (or circuit), but you should get a little bump.Cheers!

@millercarbon 
me too!

The purpose of the module type power conditioner is to clean the AC line from the receptacle to the service panel. That includes both duplexes which are connected.
 Often, multiple modules are used in separate duplexes to reduce noise on other lines which will meet at the panel. This will have an cumulative effect. 

A line power conditioner is plugged into a receptacle, it's function is to reduce noise from the mains to all components which are plugged into this unit. It is only cleaning the AC for your audio system, it does not send a cleaned return signal to the panel.

There are so many different power conditioners, each with it's own unique design.

I have multiple in my panel, step down transformer, conditioner, phono stage, amp, speakers, crossovers, several drivers, and super tweeters. 

If a modular device (e.g., Venom Defender) and a whole house device (e.g., EP-2050) can "clean" the adjacent circuits, how is a power conditioner different?