The few times I tried the "Hide if empty" option, it didn't work as expected. I assume in most or all cases it has something to do with the template; screen estate being reserved and/or styling being applied to empty module positions that contain no data. How nice would it be, if we had a JDocumentHTML::countModules() equivalent that not only determines if any modules are active for a given position, but also if any of those modules contain data. Can the "Hide if empty" logic be made easily accessible, from within a template, to achieve that?

the "Hide if empty" option should return empty for modules with no output.
That means if you have 3 empty modules in a position, the template would see this as no modules (so countModules would return false).
At least, that's the whole idea.
If it doesn't work, could you tell me what modules you are testing on? Or give me admin access...

On the current site I used it for a (sub)menu in the right column. That menu only shows level 1-N of the main menu, and is only displayed when the corresponding menu-item in the main menu is active. However, when the menu is not displayed, the right column still shows a small rectangle.

The menu's are not accessible for guest users, so you need to login to see what I mean. Where can I send the login credentials?

It might be that the module outputs an empty list. Something like:

That is not seen as empty.
So take a look in the html of the site to see if there is any html output placed by the module.