Add an option called "New Chat Message (Personal)" to the Account Settings => Notifications tab. This option should notify a user only about brand new conversations without an assignee OR about messages in conversations where this user is an assignee. Rename the current "New Chat Message" option to "New Chat Message (All)". Move the option "Has sound (chats)" from the Settings section to the Notifications section, under the New Chat message section and Rename "Has sound (chats)" to "Chat Sounds":
