The scans are directly saved to the cloud. But is there a way to save the PDF document (not the photo) to the device? I have tried opening the document in the Adobe Reader app by there seems to be no option to download the document.
Right now, there is no way to have full control over when the PDF is saved to the device (for example, for offline access). We are considering adding this feature in a future release.
A workaround for now is to select the "Share File" option, select "Import with iBooks", and the PDF will be saved to the device and available in the Apple iBooks app.