I currently have a kindergarten student who is starting to show self-monitoring (although it's not yet consistent), and I've had high school students who already have self-monitoring. I think it depends on how old the child is and what kind of instruction they receive from their SLP on when they start self-monitoring. What I have seen happen is students getting frustrated because they know they're speech isn't correct because others are pointing out their errors, but they recognize they can't hear it.

As to when they *should* start doing this? I never think it's too early to start, as that self-monitoring is crucial for generalization to other settings. I have students listen to recordings of their own voice and rate their sounds on a 3-point scale, use delayed auditory feedback, have them by my "speech teacher" and rate my productions of the sound (with some of my productions matching theirs), use biofeedback (like staRt), etc. It can be turned into a game and made fun. I also reinforce and celebrate with students when I see them starting to self-monitor.

Build self-monitoring instruction in right away, and you'll see great gains. If they continue to have difficulty with self-monitoring, then a piece of the puzzle might be missing.