Take a spell like Stoneskin. One of my BBEGs (I'm a DM) cast Stoneskin on themselves to maximize their tanking capabilities. Our party wizard asked about Dispel Magic, and if it could be used to dispel the effect. In the end, it seemed RAW, so I let him. Though later, this seemed... weird. Stoneskin technically has 3 effects (resistance to 3 different damage types) and the target was the source of the concentration. Dispel Magic removes one magical effect, not one spell.
In the future, my personal ruling is you cannot end an effect if their concentration is on themselves. The wizard seemed surprised when I let him end it before, so I think he'll agree it seems reasonable. I'm comfortable with how I'll handle this in the future. I'm just curious if there are Sage Advices or Errata covering this that neither of us have seen.