I suspect that your correspondents are using the "Inline Reply" feature that made its debut with Office 2010. When you reply inline the reply is treated by Outlook as an inline attachment, but that's not the same as a "real" attachment as we conventionally think of them.
The information at Tech-Recipes entitled, Outlook 2013: Disable the Inline Reply Feature, and at MSOutlook.info, the discussion entitled, Attachments show up in the message body, may clear things up, at least a bit. Only messages that are in RTF (Rich Text Format) support inline replies implemented as attachments.
