Saying that should be private is like saying someone's code check-ins should be private.
And if we assume people are good rather than bad, it's a chance to show off, without ego, a particularly good response and/or responder. That gives them a morale boost from knowing their boss and co-workers saw them doing a good job in a natural environment, and it motivates co-workers to do a similarly admirable side.
Fascists also wore pants, drove cars, ate sausage, drank beer, etc.
While customers may have privacy concerns and expectations regarding their interaction with a company, employee interaction with customers is not an employee-privacy issue.
Remember, employees are being paid to interact with customers. The company is liable and responsible for those interactions so it's absurd to argue that the company has limited access to those interactions because of employee privacy.
BTW - The purpose of a company is NOT to keep employees happy. Doing so may be useful, but that's different.
> "We developed our own customer support software. One of the things it does is randomly select an employee response to a customer and send that response out to the entire company and to all of our investors each day. It keeps us on our toes."
That's a clever little social hack.