Luckily, our school has Google Apps for Education. Every student has a Google account, so when they write, they do it in Google Docs. They share their paper with me and I have 24 hour access to their work. I sit down face-to-face with as many students as I can during class, but I look at every paper, everyday. For the students I don't sit down with, I insert comments into their paper. Comments are great ways to have a discussion without marking up their paper. Students can respond and you can easily have a conversation about their work. I keep track of who I talk to in person and who I comment on to keep balanced and keep track of formative progress. By the time the paper is due, I rarely need to do any "grading" and there is no need to turn anything in. Another major benefit is the "dog" can't eat it, they can't forget it at home, and since it's saved in the cloud any online device can access it. Revision history is another tool that allows you or students to recover their work in case of a problem. You can also do research without leaving your doc.
Managing all of the paper is an issue for many teachers. If you have several hundred students it can be difficult to manage that many docs. Its important to use collections/folders and be proactive in managing your files. I would also recommend teaching student about email notifications. Every time someone shares a document with you, they can send an email to notify you. If you are expecting their work, you don't want to clear several hundred email from you inbox.
The way I manage my student work is by having 2 Google accounts. I have my school account and my personal account. When I'm online, I usually operate out of my personal account and forward all my school mail to my personal account. I have students share their work to my school account, so 90% of the documents in my school's Google Docs account and my personal account isn't cluttered up by other people's work. I really don't know why every school doesn't take advantage of Google Apps for Education. Docs and formative assessment is just the tip of the iceberg.