I don't take a scrubby with a sponge attached, but use a plastic one that is more like a chore-boy, just a plastic mesh thing.  That and an SOS if we are cooking over a fire (for the sooty pots) is all I use for washing dishes.

I do take a small bottle of dish soap, but it isn't drippy and have never had a problem with it.

After scrubbing dishes the scrubby is rinsed with hot water (recently boiled) like the dishes and packed into a ziplock bag wet.  It never gets unpleasant and I have never worried about bacteria.  However, if I were to do so (and now I may) I would use the "few drops of bleach) method.  But I would not take a scrubby with a sponge attached, ever.

After a trip, a used one just is thrown away, and of course so are any used SOS pads.  I use the SOS very sparingly and often only use one on an entire trip.