For small oscillations (e.g. small "swings") the period will be INDEPENDENT of the mass of the person in the swing.
So it won't matter whether you or a small baby or some big fat dude is in the swing - the period of the swing will be THE SAME.
So, you could put different sized ("massed") people in a playground swing and have them swing back and forth. Make sure they don't swing too high and that they don't put any "effort" (i.e. extra energy) into the swing. Just have someone else push them gently in a swinging motion. The periods should be the same for anyone in the swing.
Also, the period will change significantly after a few back-and-forths because of FRICTION in the swing's pivot. So the period will only remain the same for the first few swings.
Does that help?