I wouldn't go as far to say that training barefoot is always superior to training with shoes on. There are situations where certain types of shoes provide some benefit to being completely barefoot.
In most situations though, shoes do not really provide any benefit other than comfort and sometimes can be detrimental. Really heavy, thick soled shoes (as shown in the image) in particular can be troublesome because the padding actually makes the feet a little unstable which causes the ankles to roll back and forth which creates a loss of stability. This is why flat-soled shoes like Chuck Taylor's or Vans are in-fashion gym apparel these days. Depending on gym rules, a lot of people wear Crocs these days because aside from being extremely fashion-forward, comfortable, and utilitarian, they're very easy to slip on and off for particular exercises.
Specifically for calf raises, I find that I still roll my feet in flat shoes. There is much more engagement in the calf if I do them barefoot. The difference is apparent in video. Being barefoot also gives better feedback so you can better feel how your feet move. Shoes will blunt that sensation.
So yes. It's perfectly reasonable to do calf raises barefoot if you feel better engagement and want the additional feedback.
For the safety concerns, it would be slightly better to wear a leather shoe than being barefoot. The shoe will blunt the impact a little. If dropping dumbbells on your feet is a great concern, then feel free to wear shoes. It won't make a large difference. Although, the risk is never fully mitigated. It still hurts a lot.
