Chatter posts, like any other data in Salesforce, are stored forever unless an authorized person deletes them.
Chatter Moderators Users are available in: Group, Professional, Enterprise, Unlimited, Contact Manager, and Developer Editions. For more, please read Moderate Chatter Free Users.
As an administrator, you can Assign Moderator Privileges to Chatter Free Users in your organization. You can also remove them if needed.
Note: It's not possible within standard functionality to stop Users from deleting Chatter posts completely. Also, you can use an app in AppExchange to remove permission for deleting posts. For more, please review Prevent the deletion of Chatter and Chatter Delete Blocker
Stop the Ability for Any Group Member to Delete Chatter Posts of Other Users
000385670

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