If you're experiencing issues with the "Send message" button in the Odoo chatter not sending emails from certain users, while the outgoing and incoming mail server connections are properly configured for all users, there are a few possible causes and solutions you can consider:

Check User Email Configuration:
Verify that the users experiencing the issue have their email addresses correctly set in their user profiles.
Go to Settings -> Users & Companies -> Users, select the user in question, and ensure the "Work Email" field is populated with a valid email address.

Verify User Access Rights:
Confirm that the affected users have sufficient access rights to send emails from the system.
Go to Settings -> Users & Companies -> Users, select the user, and review their access rights. Ensure they have the necessary permissions for sending emails.

Review Email Server Configuration:
Double-check the configuration of your outgoing mail server to ensure it is properly set up and accessible for all users.
Go to Settings -> General Settings -> Email -> Outgoing Mail Servers and verify the settings for the mail server being used.

Check Mail Alias Settings:
If the issue persists, check if the users experiencing the problem have any mail aliases associated with their accounts.
Go to Settings -> General Settings -> Email -> Aliases and review the aliases assigned to the affected users. Ensure they are properly configured.

Verify Email Sending Method:
Check the email sending method configured in Odoo. You can find it in Settings -> General Settings -> Email -> Outgoing Mail Servers.
Make sure the sending method (SMTP, local, or other) is appropriate for your setup and that it is correctly configured.

Check Logs for Error Messages:
Review the Odoo logs for any error messages related to email sending. Logs can provide insights into specific issues that may be causing the problem.

If the issue persists after trying the above steps, it may be necessary to further investigate the specific environment and settings of your Odoo installation.