What does the transverse sensitivity specification mean in an accelerometer datasheet?
Transverse sensitivity is the sensitivity of the accelerometer at 90 degrees to the sensitive axis of the sensor. If you were to refer to the conventional sensitivity in the sensitive axis direction as axial sensitivity, transverse sensitivity is the sensitivity at 90 degrees to that.
It is expressed as a percentage of the axial sensitivity. Ideally it would be 0%, but due to manufacturing tolerances, it can be as much as 5%. Values as low as 3% or lower are available on special request. But as the desired value goes lower, it becomes increasingly difficult (and thus more expensive) to achieve. Transverse sensitivity is sometimes referred to as "cross-axis sensitivity."
Why be concerned about transverse sensitivity?
As a user, you want to be assured that the measurement you are taking is due only to acceleration in one direction. If this were not the case, making sense of your data would be difficult, if not impossible. (Note that triaxial accelerometers are available for measuring acceleration in three orthogonal directions from the same point.) When an accelerometer is stimulated on a calibration class shaker, every effort is made to ensure the motion is in one direction, with very little transverse motion.
In this situation, you may not care that the accelerometer has a high transverse sensitivity, since the sensor does not see any motion in that direction. However, in a real test on a real structure (or even on a less than ideal shaker), we know that the motion is in all directions. In this case, a low transverse sensitivity accelerometer is crucial, as you want to be assured that the measurement you are getting is from only one direction. In this sense, the contribution of transverse sensitivity to a measurement can be thought of as a "noise" contributor to the measurement.