I own a french press.
I have found that cleaning it (screen, spring, vessle) in its entirety after each use yeilds a crisper, cleaner taste.
I can say this, for I have have made coffee daily in this press for a month with no cleaning whatsoever, just a mild rinse in tapwater. After a month the depth of flavor was incredible, but at the same time strangly reminicent of a bitter toxin.
So, my vote is for daily cleaning with a solvent that does NOT leave any taste traces (citric acid). Many soaps can leave residues that are difficult to wash off, bleaches are horrible and take forever to rinse off.
-SF