Absolutely I keep copies of all my records! I keep them on paper in file folders by body part and some types of tests, like blood tests, so I don't have too much in one folder and can find them easily. I get copies of all my MRIs, CTs, anything that can go on a disk.
I make copies of anything pertinent, including the disks, for other doctors that I see at a later date.
I also keep a journal of all my medications, complications, etc. That's how I manage to take my meds on time and keep track of what meds not to try again. I keep on my computer a list of all meds to which I have had bad reactions. (a whole typed page)
Yes, you can scan in and save everything and put it in a database if you have nothing better to do with your time. You'll still have to print stuff out for your doc visits because they don't generally look at much that's not on paper. So I don't bother with that.
Docs generally only listen to the first question asked and the first condition to which you are having a problem at the moment, so don't expect them to read your entire report, especially if it's on a disk or jump drive. They may want to compare an MRI with a slightly older one, but usually don't want the entire history, but it's good for you to have everything yourself.