The pushing database would not be available at that point. The best we’d be able to do is get the IP address that the request came from since the REST request simply contains a document to add and no other information other than standard HTTP (or web socket) headers and information. Changing this would require changes all around the board on all platforms to maintain interoperability and backwards compatibility and so is not likely to happen.
Pulling and Pushing are meant to indicate direction, and not the actions that people have come to associate with them. Pulling will only poll if it is set to continuous, and even then it keeps existing connections open until it receives information so it’s not consistently asking.
As far as bandwidth goes, the biggest source of bandwidth usage is document attachments. Everything else is pretty tight, with the overhead coming mostly from stuff from the HTTP protocol. In fact we are currently in the process of evaluating a replacement for HTTP that requires fewer connections and less bandwidth so keep checking around in the coming weeks / months if you are interested.