Both lemon juice and vinegar are acids that will help to remove dirt, sticky residues and oil from the glass. So to was the ink used by newspapers when your grandmother was young. The ink back then was less of an ink amore of a process that chemically blackened the paper. That is also what happened to the hands of any one who read the paper. A little moisture, sweaty hands or a finger lick and the persons hand and mouth would start to discolour. Anyone who accidental licked their fingers to turn the page would wind up with a black finger, lips and tongue as well as a bitter taste in there mouth. That black and bitter taste is the acid that was used back then in the printing process.
So it was not the newspaper that was cleaning the windows it was one of the different sources of acid and a healthy dose of elbow grease. The acid not only helps to remove any built up grease and grime but it also breaks down the surface tension of the water which helps with the drying process. But it comes at a cost. Ask your grand mother if using newspaper to clean windows did a good job and she will tell you a definitive YES. However if you ask her if using newspaper to clean windows left paper fibres and small rolled up bits of paper on the window and window frame she will likely also tell you a definitive YES also. So it turns out the windows were not so clean after all.
Read what Lemon juice and Vinegar can do to your windows and frames.