Should I wear shoes while performing kettlebell training?

That depends on you. Working out with bare feet strengthens the muscles and tendons in your feet and ankles (unless you have a foot problem such as plantar fasciitis that requires you to have orthotics, try to work out in bare feet). Also, your body is naturally more unstable when you are swinging the kettlebell and your feet need feedback from a hard surface to remain stable. Shoes with padding are designed to take this away and remove the body's ability to feel the ground. If you are used to wearing shoes all the time, you should gradually work up to working out in bare feet. It is not uncommon to experience some ankle soreness in the beginning from using your muscles in a new way. If you have chronic or re-occurring pain in the foot or ankle you should always use shoes for training.