And now for a different perspective..
One of the benefits of strength training is it enables people to perform better at daily activities (lifting things, playing sports, etc). That benefit is slightly less when one trains in different footwear than they perform their activities in, because the movement skills are different. For example, when you lunge in sneakers, your stabilizer muscles work harder, and your feet don't get as much "feedback" from the ground about your positioning. If you lunge often enough in sneakers, your body eventually "learns" how to do it very smoothly (as good as barefoot). Which is good, if you tend to wear sneakers for daily activities, like i do. But if you always lunge barefoot and then try to do the same movement in sneakers out in the real world, you may not perform it quite as smoothly.. and who knows, you could even get injured?
As an example - suppose a basketball player always practices shooting baskets with a ball that's twice the weight of a real basketball. When it comes time to play a real game, he won't shoot as accurately, because he never practiced throwing a light ball.
I hope that made sense. It's based on the S.A.I.D. principle: en.wikipedia.org/wiki/SAID_principle
Personally i train in low heeled shoes similar to Converse & Keds, since that's what i usually wear when i'm out and about. Plyometrics and speed drills are a lot more comfortable in them vs. barefoot.