Send fax directly to or from phone without going through a web-based service?

Is there a way to send or receive faxes directly to/from the phone without routing through a web-based service?

There are times when I've needed to send or receive a document image without going through e-mail or a third-party service. It may be a recipient without an e-mail option for a document I need to submit (e.g., a particular government agency) or it may contain sensitive information that I do not want going through other hands. When I checked the app store, everything I found seemed to involve a web-based service (I could've missed one, though).

With a device that can take images and make phone calls, it seems like a natural extension of the capabilities to be able to communicate with a fax machine directly.