Hi,I've stayed at a few hostels in Japan, and a private room is exactly that... your own room with own key or access card, etc.
I always opt for private rooms where possible, and if the hostel has single private rooms (often, if they have private rooms, they're twin or doubles), they're normally not *that* much more expensive than a dorm... or at least I don't think so.
Generally hostel dorms will at least have a small locker where you can store valuables, but confirm this with them first, if you're particularly worried. That said, I found in Japan, tourists tended to adopt the courteous Japanese attitude, and I feel it would be unlikely things would go missing.
One guy who was staying at my hostel would leave his laptop lying around all over the place, while he went out for the day, and no one ever touched it.