I'm about to get rid of an older scanner and I want to make sure that no one is able to acquire any sensitive information that was scanned previously with the machine. I assume it does have some kind of internal memory. How long does it store scans? Until I turn the scanner off or unplug it? Can I check/erase it? Is there a hard-reset button on the machine that resets to factory defaults?
Basically, no, flatbed scanners don't remember old scans.
A flatbed scanner sees a lot of incoming data (the pixels) and outputs them all on whatever communication medium it has (say, USB). RAM is used as a buffer between the two. Cheap scanners don't have a lot of RAM, much less than enough to contain one scan, and rely on the host computer to be able to read the data fast enough (or to stall the scanning in case of overflow, although this would be likely to induce picture distortion).
Network scanner usually have more RAM, because they may need to contend with network congestion, so they must be able to buffer one full picture. Some may buffer several pictures, in particular automatic scanners coupled with a printer (so that you may efficiently copy a bunch of sheets). None of them, though, has any business storing files permanently so it would make little sense for them to use anything else than RAM. Correspondingly, after a few minutes without power, the data is gone for good.
Some advanced scanners who act as file servers (so that network machine may obtain scan results afterwards) could store data permanently, and require special procedures. I really doubt your scanner would be of that kind, though. It would not be reasonable in terms of economics.
It depends upon which type of storage is used for image buffers. If I were designing it, I'd use volatile RAM such as CMOS SRAM, which would quickly lose integrity upon power loss. That being said, some devices do use NVRAM (e.g. EEPROM) for scan buffers in order to maintain buffers if any problems occur, or even to store images for subsequent downloading over USB.