When a new Chatter post is made in a case's feed from within an Experience Cloud Site, users who have access to the case will not receive email notifications.
When a user responds to a case from within an Experience Cloud site, only the case owner is notified by default. Other users who have access to the case do not receive email notifications for Chatter posts made on the case feed.
To ensure the case owner receives notifications when a comment is posted on a case, navigate to Setup > Feature Settings > Service > Support Settings and confirm that "Notify case owner when case is commented on" is selected. Click Save.
The only way for users other than the owner to receive email notifications when a Chatter post is made on a case is to create a Flow that sends email notifications to those users. Contact your Salesforce administrator to implement this automation.
When the owner responds via a comment, the user who originally responded to the owner's post may receive an email and notification depending on how they have configured their personal notification settings. See the Additional Resources section for guidance on configuring personal Chatter email settings.
Guidelines for Managing Your Personal Chatter Email Settings
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