Users will not see a copy of messages which they sent to group lists if they are using Gmail or Google mail services to manage their email. This is an intended 'feature, not a bug' benefit, courtesy of Gmail - it is not because Kavi doesn't broadcast the user back a copy (as a member of the group list), but is a result of Google hiding it automatically. This is explained in Google's help documentation:
When you send mail to any group or mailing list you subscribe to, Gmail automatically skips your inbox and archives the message to save you time and prevent clutter. The message will appear in your inbox if someone responds to it or if there is an error delivering the message. If you'd like to view your message, you can find it in Sent Mail or All Mail.
Remember that you also have the ability to check the web-based email archives for any Kavi Group to verify that the mail was, in fact, sent to the Group list as well!