Got my Vivo Barefoot pair a couple of days back. Have had a run and a couple of walks :) so far. Very very comfortable.
That grey shoe-like thing lying beside the shoes, is the inner-sock with a sole. This inner-sock makes the shoes 'puncture-resistant'. But the shoes can be worn without this also if one likes, especially in watery conditions I guess.
In this pic right above, I've worn socks as well. But they look sexy without the socks actually. My fascination with red continues. Head-turners surely :).