Revisiting this old issue.
One of my biggest issues with chat is that after just two minutes, not only can we not edit messages, but we also cannot delete them. While I am not advocating for an unlimited delete feature, I'm suggesting a compromise that allows us to delete (but not edit) messages that are more than two minutes old, but not indefinitely: 5 to 15 minutes would be sufficient.
To protect against abuse, this would be balanced against a limit of two to five such deletions within a one- to four-hour (or similar) period, and/or an autoflag if a user makes excessive use of this feature.
I realize that this will add complexity, but the alternative is currently to flag for mod attention each time someone erroneously posts personal information and it needs to be removed. Quite often, this happens shortly after the two-minute edit window has passed. The mod flag safety net would still be there for those edge cases where an older message indeed needs to be removed.