This centre of mass will also be mobile as the moon revolves around the earth.
There we have a problem, since the natural frame to choose is the one in which the center of mass is fixed and both bodies mutually orbit around the center of mass. That way you see the motion of the tide-susceptible parts of the Earth clearly.
Thus, a person standing on the earths surface will be moving in and out relative to the centre of mass of the earth/moon pair and will experience (very tiny! 2micro m/s2) variations in gravity.
Only because the Earth is rotating at a different rate to the Earth-Moon system. On the Moon this is not the case, since the Moon is tidally locked, yet there are still tides raised. That discussion of in-out motion also jumps right past where the tides arise.
They would also be moving in and out relative to the L1 Lagrange point, ie varying their range from a point of zero g
L1 does not have zero gravity. It has a zero gradient of gravity. If it had zero gravity, things would not be able to be kept orbiting there, but would instead drift off in a straight line.