Yes. Anything personal, confidential, etc. Right now I don't have much, but at my last job one of my coworkers would draw backgrounds for my computer and I took those with me, as well as photos I'd taken and put on my computer for backgrounds. Also documents I worked on that weren't requested by my job (explanations of diagnoses, for example).
Yes. I have some files on my personal drive. Nothing really important or super personal, mostly some rescue paperwork that I saved their, and some photos.