I think this is a common feature because I bought a cheap printer from curry's a few weeks ago HP deskjet 3070, and it has a scan to computer feature, where you scan on the printer panel and choose which computer to save it to
Pretty much most of the "networked" HP all-in-ones will have a menu option to do that.
You would choose the "Scan to Computer" option on the menu screen of the printer, select the relevant computer from the list and have the HP scanning application open itself on that computer with the scanned image.
More expensive models (ie: Officejet Pro 8600 Plus) also have a so called digital sender function. This is commonly used in large offices to digitise large volumes multi-page documents with minimum effort. It will create pdf files and save them to a networked hard drive/network share on a computer - all without involving a computer to process the data.