These apps (including PaperShip) can sync with Zotero using ZotFile or the browser version of Zotero on your tablet (see the Mobile Devices tab on this guide). Note: this listing does not include tablet apps such as iAnnotate, PDF Expert, and GoodReader (all iOS) that enable you to annotate PDFs and other documents.
These tools will allow you to do this (and you can always print out a clean copy of the original PDF (without markup - annotations, stamps, etc.) if you need to). Sometimes you may want to write directly on a PDF in addition to creating notes about it in Zotero.