"He cut the paper with precision": "with" is being used to indicate the tool or method used to perform the action of cutting. In this case, "with" is being used to show that the person cut the paper using precision.
"He cut the paper with me": "with" is being used to indicate the company or accompaniment of the person involved in the action. In this case, "with" is being used to show that the person cut the paper while being accompanied by someone else (i.e., the speaker).
"He cut the paper with scissors": "with" is being used to indicate the tool or instrument used to perform the action of cutting. In this case, "with" is being used to show that the person cut the paper using scissors.