I've always been a great believer in getting pupils to assess themselves. I think being able to assess yourself is a critical skill for making progress, and the research on assessment for learning bears this out. However, there are some pitfalls that can trip up the unwary. I'll go through what they are, and then suggest how to avoid them.
The most common problem I've found with self-assessment is that pupils tend to undervalue what they have done. For want of better terminology, they might declare their work to be good, while I might declare it to be very good. I don't know if this is peculiar to English schools, which are the only schools I've taught in. It is certainly the case that, traditionally at least, the English have tended to be reluctant to blow their own trumpet or, as my American friends tend to put it, toot their own horn. I'm not sure that is quite as true as it once was, judging from the number of "humble brags" on social media, but still. That has been my experience at least.
This is less common, but sometimes pupils give greater weight to a piece of work than it really deserves.
This can crop up when asking pupils to submit examples of work for an eportfolio or something of that nature. It may seem odd that someone might forget to mention their best work, but it does happen. Here are two examples I've come across.
Example 1. Some years ago I was putting together a bid for some Government money for use in schools, so I visited the headteachers concerned and asked them a series of questions that would enable me to put the bid together. Here's what happened in one of those interviews:
Me: What provision do you make for children with special educational needs?
Headteacher: Not that I can think of?
Me: So is it not the case that you have a special unit for deaf children, and that its work is admired all over the borough?
Headteacher. Oh yes. I'd completely forgotten about that.
Example 2. After I had told someone what sort of things I do to earn a living, during a break in a conference, I realised that I had completely forgotten to mention the courses in assessing computing that I run. (There's some information here if you're interested.)
Remembering achievement but underestimating its significance
For example, a pupil may remember that she has written a computer program that works, but completely fail to appreciate the significance of the fact that she incorporated concepts that have not yet been covered in lessons.
Misinterpreting the task
In some cases, pupils may do some work which, in itself, is brilliant, but which is also completely irrelevant, because they were asked to do something else. Unfortunately, as my years working as an A Level examiner confirmed, you can't give credit for work that wasn't asked for, no matter how brilliant it is.
And if this sounds harsh, then all I can say is that first, it's the way life works. If your headteacher asks you to write a report for parents on how well the new Bring Your Own Device scheme is working out, and you write a brilliant report on how the new Computing curriculum is panning out, I doubt that you will receive any thanks.
Second, I am firmly of the opinion that being "capable" (however defined) in a subject includes understanding the reequirements of what you have been asked to do.
My own experience, which is borne out by research, is that you can't introduce self-assessment just by telling the pupils to get on with it. You have to put a number of things in place:
Make the criteria clear
Nobody can expect to assess whether they have done well or not if they don't know what they are making the judgements against. The assessment criteria have to be made clear to pupils from the outset.
One of the ways I've always done this is by putting the criteria into pupil speak. Another is to prepare examples of pieces of work that have been marked against the criteria. Yet another is to get the pupils to mark someone else's piece of work -- see the next point.
Use peer assessment
I've found that pupils tend to be more forgiving of their friends' work than they are of their own, so this can pose a problem in itself. So I think it's important to not make grades explicit in any rubric or marking scheme you give out, only the criteria.
Notwithstanding this caveat, peer assessment is useful for several reasons, not least of which is the fact it involves having a conversation about how well a piece of work meets the assessment criteria.
Link achievement to goals
If the aim of a piece of work (or a subsection of it) is made clear, then the question for the pupil becomes not "How well have I done on this task?" but "How far does what I've done meet the objective?" (Obviously, expressed in terms they will understand.)
Go through Examiners' reports
If you teach pupils taking GCSE or A Level, it's crucial to spend time going through the Examiners' Reports to find out what common errors pupils made when taking the examination. This is clearly difficult to do when, as has happened in the realm of Computing in England, new specifications have been introduced and your pupils are taking the exam for the first time. If that's the case, reading Examiners' Reports for the old qualification can still be useful, especially if you have kept with the same Awarding Body, as is reading all the guidance available and attending relevant meetings where possible. The aim is to get inside the head of the examiner so that you can pass on appropriate information to pupils.
Self-assessment is a useful tool for pupils, and can help the teacher understand how a pupil is thinking. But it's not a skill we are born with, so it's important to set up the conditions necessary for it to succeed.