It seems to be SHIELD backed, by home controlled. Such as, Rhodes apparently is in charge of the base, which was given to him by Tony, but that might've happened while Tony was still in charge of the entire SHRA Initiative before he took the head of SHIELD over. As such a base would already be in construction, and people being handpicked already.
It seems the Initiative itself though is mainly based on Homeland Security, or a new branch of the armed forces, dealing not with military issues, and can only function on it's home turf (as, unlike Ultimates), they haven't been shown to have any interest outside of it's borders.
But this is odd as well, as it seems the American Initiative seems to be helping out other countries with their own form (iIrc), where it should be SHIELD doing that.
It seems that SHIELD simply has influence, but not control.
So, in truth, it's like the real UN. It likes to make out like it makes a difference, then turns around and makes no difference.