It is there as a vibration damper. Not usually available separately, so once they let go, they're 'gone'. If it is detached, it is more likely to cause a vibration than subdue one, so hook it off. You may feel some harshness in the drive. If so, this will eventually affect bearings and the driveshaft joints, reducing the life of the driveshaft. Urgency for complete shaft replacement will depend on annoyance factor, if any vibration is there at all.
The left one does not have it, as the shaft is shorter. The shorter shaft is also usually a larger diameter, to make them equal mass, helps reduce torque steer. Lots of strange physics going on, but if a new damper is not available, you can't fix it.