Michael Pascoe has a video on Bioscreencast that demonstrates how he gets PDFs into Papers from a web browser. It involves going through the Print... panel and choosing "Save PDF to Papers", which creates a brand-new PDF and then imports it into Papers.
The problem is that creating a new PDF removes all the metadata that was in the original document. Given the current state of publishers' metadata there probably wasn't much there anyway, and Papers has the nice feature for searching elsewhere (eg PubMed) then importing the metadata and matching it to the PDF.
Perhaps Papers could offer to import the original PDF (through a Services menu option?), but either way it would still be useful if Papers could look through the PDF, check for a DOI and use that to look up the basic metadata from CrossRef or PubMed (you can search for a DOI in PubMed and get the PMID, then use that to look up more metadata).
The other step is getting the metadata back out of Papers and into a reference/bookmark manager like Connotea or Zotero. I guess synchronisation through APIs and a remote server is a good way to go for this, but it would be nice to have a local method (involving Applescript or the Services menu again?) that would open a file containing the bibliographic metadata (RIS/BibTex/MODS/RDF) for the selected item in your web browser where it could be captured by your default handler application.