Should I do something about a staff member using private or public paper to photocopy a university document?

There is no harm in photocopying the paper, because the original assumption is that the person who photocopied it was authorized to do so. Also, the jurists have stipulated that a trustee is permitted to dispose of the property of the one who entrusted him with it in a small way that is customary and for which there is no contention. They cited the verse: ﴿or what you possess the keys of, or your friend﴾. Moreover, photocopying a single paper is considered a trivial matter that is overlooked, as Ibn Abd al-Barr mentioned that a person may eat from his brother's property something trivial that pleases him. And Al-Nawawi mentioned in his commentary on Sahih (definition) Muslim that it is permissible for a person to enter another's property without his permission if he knows that he would agree to it, and that this is not limited to entering land but also includes using tools. Therefore, there is no need to be scrupulous about this matter.