Got a response from Nokia. It turns out that the screenshot sound CAN be turned off although I still don't understand why it doesn't respect the 'vibrate mode' setting. They seem to think that you can turn off the 'charging stopped' chime as well, but it doesn't work for me. Full response below.
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Regarding your concern about your Nokia Lumia 920's silent settings, we would like to inform you that you can disable everything such as the chime when charging, screen shot shutter, even in the sound of when connecting it to the charger as this is not a fault of the phone. To do this please follow the steps below:
1. Tap on Settings
2. Select ringtones+sounds
3. Turn off the following:
a. Ringer
b. Vibrate
4. If you want everything to be silent please select none on the following fields:
a. New text or IM
b. New voicemail
c. new email
5. If you want to turn off all extra tones on your phone, please un-tick the following:
a. Reminders
b. Key press
c. Lock and unlock
d. Camera shutter
e. All other notification
Please note that even if you follow the steps above, the alarm will still sounds on the specific time.