It's well known that the near side of the Moon and the far side have different topographical features and that if the Moon had a core it might have been shifted to the near side that faces Earth due to a large impact it might have sustained in the past, but do we know which side is heavier? I could guess that if the core was shifted that might make the near side heavier, but I haven't found a definite answer. Please attach any evidence that supports your answer!
[...] current understanding of the structure of the Moon involves an iron-rich, aluminum-poor mantle whose center of mass is offset about 10 km from the center of mass of an aluminum-rich, iron-poor crust. The direction of offset is toward the Earth, about 14 degrees to the east of the Earth-Moon line. Such a model accounts for the basaltic maria which face the Earth, and the aluminum-rich highlands on the Moon's far side, and for a 2 km offset between the observed center of mass and center of figure for the Moon.
Will, https://arxiv.org/abs/1403.7377 , p. 51
According to this description, the average density of the half of the sphere that is closer to the earth would be greater, because it would contain more than half the volume of the denser mantle, and less than half of the less dense crust.
According to this NASA page:
The mass of the Moon is not evenly distributed; mass concentrations, called Mascons, lie beneath many of the lunar basins, and the center of mass of the Moon is displaced several kilometers towards the Earth.
This indicates that the near side is slightly heavier.