You can do cross-domain POSTs. You just cannot get a response from the server. Often times though you can get the length of the response. This can be used to do a POST to change settings on an unsecured Wi-Fi routers, for example and checking whether the call succeeded or not by the length of the response.
Everything their SDKs do can also be done using their REST API (https://www.dropbox.com/developers/reference/api) and they support JSONP to allow cross-domain GET queries. However, my understanding is that JSONP is not supported for other REST methods, most importantly POSTs. A person claims to have found a workaround for doing POSTs via JSONP here: http://stackoverflow.com/questions/5345493/using-put-post-de....
Has anyone tried that or know of a way to do cross-domain POSTs?