By default wp_list_categories() (and the widget built on it) will ignore private Posts ...
Are you saying that I should look at changing wp_list_categories so that it doesn't ignore them?
I don't quite grasp how ignoring Posts has the affect of insuring that the resulting list contains information. I would think that ignoring Private Posts might be an effective way to prevent non-logged in users (the Public) from finding out that such posts which are not visible to them exist. This sounds like a good idea which would be consistent with the notion of privacy. However, failing to remove indicators that invisible Posts exist is a minor discrepancy when compared to what I am experiencing.
I've also found some more inconsistencies which ought to be troubling to site operators. When you select "Categories" on the sub-menu for Posts on the Dashboard you are shown a table which lists all the categories with some columns which contain information about them. One of the columns contains links that show the number of Posts. When you select that link all of those Posts are displayed in a list. In my case the link says zero (0) but you can still select the link and see all of the Private Posts. Is this contradiction a design objective? Likewise, as I previously stated the pages with a Category Widget that says that there are No Categories shows Categories on the Posts and if you select that link you get a page that lists all of the corresponding Private Posts. Is this the way to make users confident in the accuracy of the Category Widget?
It is not intuitively obvious why the inclusion|omission of Private Posts cannot be dependent on the visibility that is in effect. In that, ignore Private Posts for Public (non-logged in) Users but include them for Private (logged in) Users. In that, in the same manner used to determine which Posts are visible.
I'm thinking I should get rid of the senseless Category Widget and use a Sticky Post which is assigned to all of the Categories. Then I get what the Category Widget should be producing but in a somewhat less elegant manner.
I'd also like to point out that the same logic defects seem to pertain to the "Recent Posts" and "Recent Comments" widgets.