Agents or Customers may notice that Auto-Scrolling doesn't apply to a chat window while scrolling up through past messages. This is currently working as designed within Salesforce. When a new message comes in either on the agent or the customer side and they are not at the most recent message at the bottom of the chat window or widget, the chat will not auto-scroll to this new message. Instead, there will be a notification that a message has come in.  The agent/customer will need to scroll down the chat conversation to see the most recent message.  

Utilizing Omni-Channel or Live Agent Routing Methods will have sound notifications if they are enabled for your configuration. This is for the agent and the customer side of the chat. Agents will see their tab light up as well to signify that a new chat has come in. If you are using Snap-Ins, the customer will receive a notification in the chat widget that new messages have been received.

Note:-  There is no exact document available for this.

An Idea is also raised for this similar query,

https://trailblazer.salesforce.com/ideaView?id=0873A0000015CfmQAE

I hope you find the above information is helpful. If it does, please mark as Best Answer to help others too.

Thanks.