Should I bond a water softener's in/out pipes?
It is important to understand the reason for grounding/bonding the water pipe system in a house.  The electrical service panel is typically grounded via an 8 ft or more ground rod at the meter and/or panel AS WELL AS via a bonding conductor to the metallic water source pipe. (Plastic feeds are not bonded.) The grounding wire is typically #4 copper. 

The grounding of the water pipe is not necessary for the proper grounding of the electrical system if there is a separate earth ground supplied (ground rods, etc.), however it is necessary in order to assure an equal ground potential between the electrical system and metallic plumbing system. 

The reason for having both the electrical and plumbing bonded at the same potential is to eliminate any possible voltage potential between the two when electrical appliances that contact water are used (electric water heaters, ice makers in fridge, garbage disposals, etc.), and to deal with any accidental contact of voltage to the water system. If this was not done, it may be possible for voltage to appear on the plumbing fixtures. 

Regarding your first question about bonding the two sides of the water softening system, the recommended bonding is to assure the upside metallic system is at the same ground potential as the downside piping, for the reasons described above.