Can students self-assess? Should we ask them to?

The Teaching Professor suggests that they can and we should:

Can students accurately assess their work? Most of us would say no with some conviction. But could they accurately evaluate their work under conditions that separated the grade they’d like to receive from the one they think their work deserves? A study in Great Britain found that they could. Even more surprising, the 160 students in this sample were first semester college students. The researcher asked them to estimate their grade on completed work using a 100 percentage point scale and 60 percent of them were within 10 percent of the grade given by the teacher. Equally surprising was the fact that when students were not within 10 percent, under-evaluation occurred more often than over-evaluation. Almost 60 percent under estimated their grade.

...However, other research has shown that students are quite mystified as to the purpose behind teachers’ requests to self assess. They don’t understand why the teacher who has complete control over the grade would ask them to evaluate their work. Teachers need to explore with students the role of this skill in professional contexts and then design activities that give students the opportunity to practice and develop the skill—which is not the same as asking them to “grade” their work.

This has me thinking about the writing class I taught this past spring. Among the problems I had was explaining to students a) the value of revising their work and b) how to truly revise their papers, as opposed to simply fixing the typos and grammatical errors that I pointed out in their first drafts. I wonder what would happen if I asked them to assess their first drafts by asking them:
1) what grade do think you will receive?
2) what grade do you think you deserve?
3) If there is a discrepancy between your answers to #1 and #2, please explain. [My prior is that the grade they think they deserve will be the same or higher than the grade they believe they will receive so their answers may reveal something about what they think of how I grade]
4) Given additional time, what could you do to improve your paper? [Although I know that some students will say they don't know, I assume that at least a few students will be able to recognize that their papers could be better, which should lead to a discussion of a) why didn't they do those things in the first place and b) how to incorporate those things into their revisions]

I have never had students do this sort of self-assessment before so I have no idea what the results might be...