If you want to discuss the Fantastic Widgets shop and you know you both do the shops there is no reason to mention the MSC at all. And since, as is the case with the fatherly pizza shops, some clients are shopped by multiple companies, you can't automatically assume that the person with whom you are in conversation is shopping Fantastic Widgets for the same company as you are.
It really boils down to doing your best to honor your agreements. Jacob and his moderators do an excellent job of trying to MAKE us be compliant to our agreements in open forum because yes, there is the tendency when discussing the Fantastic Widgets to say "I wish SuperBIG MSC would drop that . . ." But ultimately we are the ones responsible for honoring our agreements. And you have been around long enough to have seen shoppers chastised by MSCs for posting too much information and those shoppers rushing back over here to edit their posts. And 'too much information' varies a lot by MSC. If I recall correctly, for one it was a disclosure of fee, for another it was too much information about the questionnaire, etc.
If a scheduler has loose lips about the shops of companies other than they one they are scheduling you for in that particular interaction, you should probably wonder about their ability to keep YOUR information confidential as well.