I had a high-school student ask me this question, and quite frankly, I don't know the answer. It seems obvious by drawing a free body diagram that the gravitational force vector on you, if you are standing on the side of the earth facing the moon, would be smaller than that if you are standing on the other side of the earth. But what is really confusing me is whether or not you could actually MEASURE this effect using a scale (even assuming perfect precision). It seems to me that you could measure it. But I'm not sure. Can someone please clarify?