We have a no “street/outdoor shoes” policy. Yoga and pilates classes are the only ones done barefoot. I do, however, sometimes have my clients train barefoot if we’re doing balance training or other full body exercises where no external weights/equipment is used; the feeling of being more closely rooted to the ground is very helpful and empowering.
With regard to the “five finger” shoes, my understanding was that they were designed to be used in lieu of barefoot i.e, in yoga, pilates or some other gym activities. I agree they can be dangerous if used in place of a regular athletic shoe, when lifting weights for example. You would think common sense would reign supreme when people are deciding what the best footwear is for a particular activity; obviously this is not the case. I guess each facility has to state its policy clearly, and ensure that all users are aware of it.