Will depend on the scanner used and the custom setups the hospital use.
Some scanners print some data onto the electronic image. What is depends on the scanner/setup and can’t normally be changed. However, if the images are DICOM (they almost always are now), selected DICOM data can be written to the hard copy. Whether the person printing knows how to/is prepred to do this. The DICOM data is included in the image headers and is editable, unlike the scanner data which is part of the image if that’s what you mean by “hard printed”
EDIT – just looked at he other thread. Normally we don’t provide hard copy images, we provide images on CD. These are DICOM images and the headers with name, date, institution, etc are readable although not normally visible. If can get a CD, reading the headers is easy