Ok, with 'Don't display results until filtering' set, the two problematic portlets displayed correctly, without data, as one would expect. Upon filtering, a new error message was generated:
Before, we were getting just "ERROR Contact System Administrator." Looking into logs, now, to see if the message in the log is any different, more informative.
Prior to filtering, did not see anything in the selected parameters different from my login, or the login of others, where this is working. Also, we had a DB restart at noon, related to a unrelated issue – and we had no luck as the restart did not help us with this issue (but it did fix the issue which called for the restart).
I then logged into Clarity on the user's machine, same browser, using my login – and, everything worked fine, no errors.
Will get the error log, next. Also, expect we will run an SQL trace with the user logged in on his own machine, once he is free (Friday?). For now, that's what we know.