I agree with this, but I feel like a normal user shouldn't be able to. Maybe whatever the user rank is with 50+ posts, or somewhere around there. There are general reasons why we don't want people to edit the shoutbox, an example being someone insulting another member, then editing their post to basically compliment said user, hiding the insult from forum mods/admins.
That's why I say 50+ posts, that way it's more of a chance of a trusted user only editing it.
I will say, we do need to be able to edit it though, that way if we make a mistake/type in the chatbox, I want to be able to fix that type instead of having to post another chat correcting it.
Editing/removing chats helps with moderation, plus helps keep the chatbox clean of spam.
Or, at least allow 50+ users to post, then have moderators that can remove chats, mute people, etc.