The basic story seems to be that yes, under Nick Fury, it was an international organisation. However, the US provided enough resources to it that it felt it should have a say in what SHIELD did. When Nick left, Maria Hill was appointed over more-qualified candidates on this basis, to make sure that SHIELD toed the US line. Then we had Civil War, Director Stark, and presumably the US disbanded it, possibly by just saying something like "okay, no more funds for SHIELD. And all that stuff our money bought? We're taking it as well." Haven't seen any evidence of anyone besides the US making an official claim on SHIELD leftovers, at any rate.
And whether HAMMER is a-supposedly-internantional-but-really-US-entity like SHIELD was near the end, or a purely american entity, I'd guess the latter, from how it's been shown so far.