Browser not showing notification for incoming Chat

Since users can control whether they want to receive these notifications or not, if they do not show, you would need to check the browser settings of the concerned user.

For Chrome:

1. Click the vertical ellipsis icon in the upper right part of the page.
2. Click Settings.
3. Click Privacy and Security on the left-hand menu
4. Click Site Settings.
5. Click Notifications.
6. Toggle the switch icon to On.

For Firefox:

1. Click the 3-bar icon in the upper right part of the page.
2. Click Settings.
3. Click Privacy & Security on the left side bar.
4. Under Permissions, click Settings beside Notifications.

For Edge: 

1. Click the three-dot menu in the upper right.
2. Click Settings.
3. Select Cookies and site permissions.
4. Click Notifications.
5. Ensure the Salesforce domain is not blocked.

Might these settings be greyed out, your IT Department might have blocked them.

Note: Legacy Chat (Live Agent) is scheduled for retirement. Consider migrating to Enhanced Chat (MAIW). See Additional Resources.