Calgon's a prevention rather than a cure, so running it through your pump after a day's work would do nothing. You need to add it to the water you clean with, e.g. a quarter of a tablet per tank full in the porty or something (just guessing, don't know what the amount would be). All it does is soften water, helping to minimise scale build up.
If you live in a hard water area you can keep a portable ticking over by running descaler through it every month or two. Stick your solution hoses on to the machine with an open-ended hose on the end, add a mix of hot descaler to the machine and circulate it back into the tank for 10 minutes or so (run it through your wands and tools as well). Let it stand for an hour or so, heat and circulate again and then run it off into a container. You'll be able to use it again, maybe top it up with some fresh descaling product each time
Flush the system out very thoroughly with plain water afterwards!!!
A diaphragm pump ( normally the lower pressure ones below 200psi) shouldn't really suffer from scale because there's no metal parts the water touches to cause scale deposits. I've never had a porty with a piston pump so don't know if they're any different
Normally it's the heater element in the machine tank that needs descaling, all that will happen if it scales up is it will take a bit longer to heat the water, and maybe in the case of severe scale will blow the heater element after a very long time because it'll overheat itself because it can't lose the heat quick enough due to the limescale jacket it's wearing!
If you use an inline heater, much more important to keep scale-free as it will badly affect heating and also flow rate through the heater with severe build up.
If you need a descaler, search for Phos which is made by Clover Products
But going back to the original subject, DI water might not make any noticeable difference to the cleaning results but will be kinder to your machine!