Up until last week, moderators would have to contact the Community ManagersCommunity Managers to redact private information from a post, but thanks to JarrodJarrod this is now been turned on for moderators to handle.
By going to the post revision page, mods will now see a redact option:
Choosing this, you'll be provided with the standard editing controls where you can remove the sensitive details. Once you've saved the changes it will go into the moderator queue as a new flag for review by another mod. This is so there will be some sanity checking before the redaction takes place. When the revision takes place, the original revision is redacted and a special PostHistory entry is created to store the original version and serve as a record that the redaction took place. Community Managers can see a list of all redactions and review the changes made if needed.