I found you can actually add it to the ticket ‘overview’ you can show the customers in the web interface, via Manage > Overviews > My Tickets and then selecting the ‘Owner’ field under ‘attributes’.
However I did not find how to make the field visible in the ‘customer ticket zoom’.
it might be ‘protected’ somehow but it definitely is not ‘secret’ as it is sent out in the JSON response to the customers’ browser 
I understand that as a customer, you shouldn’t need to be able to pick ‘favorites’. So it does not make sense most often for the customer to be able to assign the ticket to a certain owner.
But depending on the type of organisation you work in, it might be desirable to show the assigned ‘ticket owner’ in the UI: so if you decide to for instance call about an open ticket, you can say: I’m calling about ticket #42, it’s assigned to Sarah, is she available? Or whatever. Also, it makes the experience for the customer a little bit more ‘human’, if they can see that their ticket is assigned to this name they might ‘know’.