Privacy or security concerns?
No these machines do not save any information. So don't worry about any thing.
Without working for that store you used, you cannot say that with any certainty. The machine itself may not store anything but what about the store's computer system that it is located at? They may have their own protocols for keeping a log of photos printed. If you are afraid of your photo being stored in the system of a print service, print your photos at home and never show them on the internet or send through email. Because email and attachments can be stored in mirror servers and be hacked into by hackers. So never send any attachment through email that has private info like SSN or credit card numbers because they can be read and stolen by hackers.
Most internet provider email servers are not secure unlike private business email servers.