It took some time, digging and soul-searching, but it turned out to work fine for all but me. The reason: NoScript! My favorite Firefox Addon has, so I learned, “Automatic Secure Cookie Management” as a countermeasure against HTTPS cookie hijacking (by setting cookies “secure” if they’re set in HTTPS and if they contain something resembling a session-id?). And that feature indeed can break stuff.
So if you’re using NoScript and you’re running into weird cookie-related problems: try with “Automatic Secure Cookie Management” turned off, or add the site you’re on as an exception and you might be good to go.