There are distinct schools of thought regarding "barefoot" training and "barefoot" running. What is your philosophy about it?

I have found barefoot training an instrumental component for strengthening my feet and lower body. My running form has become stronger and easier. As for barefoot running, I agree with Jason, it is does require a mindful progression from your current running shoe to a barefoot or minimalist running shoe. One of my client's trained for and ran a half marathon this year wearing the Five Finger Vibram's (a minimalist running shoe). He wishes he could wear his FFVs all the time. With a slow and steady build up of running barefoot along with the proper foot strike, my client is now a successful barefoot runner.

Aside from the safety of the environment where we run, I think it is good to remember the physiology of our bodies. We aren't born with shoes. Our bodies are made to run, and our early history is of running through the lands (mostly barefoot) to chase our food or to avoid becoming food. A healthy body is natural and so we a healthy body doesn't need shoes to support our ankles or knees.
At some particular time we did begin to wrap our feet to protect from cold elements and surface pain. That is because our bodies also have a brain!
What I hope for all of us is educated and compassionate support for our clients. I suggest we give them evidence based material on both sides of the debate and then support their decision.

Except for when I'm using very heavy weights, I like to train barefoot. I love the feeling of stability and "rootedness" it gives me with the floor/ground in moves like squats, plies, lunges, balance work and the lightness it allows for movements with the foam roller.

I'm no longer a runner, but I see no problem with barefoot running. All this "new" trend talk and examination into it is a bit ridiculous. Look back in human history, there were no shoes! That said, I wouldn't necessarily encourage a newbie to take off for her first run barefoot! Some thought, preparation, and common sense should prevail.

I believe moderation is so important.

There are obvious benefits to training in one's barefeet. To this day there are many cultures who do everything in barefeet. Their environment is conducive to living in barefeet.

This is not the case in NYC. My preference is to be barefoot at home for now. My feet are my livelihood and I have become accustomed to wearing supportive shoes. My feet like it.