Issue View of the Chat
In addition to using Chats Page to respond to the customer as a support person/agent, you can use Jira issue pages for that purpose, as you would normally do for other types of service requests - from your point of view chats are just regular Jira issues:
All messages from the customer are shown as issue comments and the files which they attach to the chat are also shown in separate comments (adjacent comments from the same author are concatenated together if you check the "Merge adjacent comments from the same author"). If you check the "Automatically refresh this page on new customer messages", new comments are shown automatically in the page, without any need to manually refresh.
When you respond to the customer with a comment, they see it immediately as a message in the chat. When you resolve the issue, they also get notified of this fact:
If an issue originates from a chat, its view gets an additional side panel, providing some additional details of the conversation:
The most important information on this panel is probably whether the customer is still online or not.
By clicking the 
icon next to the chat user name, you can read extended information about the chat user, such as:
- page where the user chats from
- their IP address
- geographical location (country - based on IP address)
- language
- time zone
