Short answer
HubSpot's KB: notifications go to the HubSpot app in Slack by default, automatically when your HubSpot and Slack emails match. For a private channel, add the HubSpot app to it, once per channel. Associate a Slack channel with the inbox to get live chat notifications and reply to visitors in the Slack thread.
1. Notifications
Notification settings in my demo portal.

2. Email matching
If notifications don't arrive, check that your HubSpot and Slack user emails match.