Answer
Open the conversation's More menu, either on its row in Inbox or at the top of the open chat, and choose Assign To. Pick a teammate, or choose Unassigned. A check marks whoever is assigned now, and your choice saves right away.
The small photo on the corner of the customer's picture shows who is assigned. On a computer, hover over it to see their name. Screen readers include it with the conversation, for example Assigned to Sam.
The chat gets a short note that only your team can see, such as Assigned this conversation to Sam. The customer doesn't see the note or the change, and assigning doesn't send a notification. After that, new customer messages in the conversation notify the assigned teammate. When nobody is assigned, they notify your team.
Only Inbox conversations can be assigned this way. Moving a conversation to Inbox, including with Switch to Live Support, assigns it to you. Conversations in HelperHat, Done, and Trash keep their last teammate, and search finds them by that teammate's name.