There are lots of different approaches you can use for this.
For example, I worked with one team that used custom fields to show the individuals that were involved in a ticket (e.g. "Developer", "Tester"). They populated the custom fields using pop-ups during transitions. This saves people from having to trawl through the ticket history, but it can get a bit confusing if a ticket is re-opened.
Another team I worked with would use a default assignee for each stage of the process. The default assignee would take responsibility for the ticket until they either did the work themselves or handed it off to somebody else.
My advice would be to try an approach for a few weeks and see if it is working well. Focus on the value you are trying to get from the visibility of assignees.