Since suspended users are unable to tell their side of the story on meta or chat, the less said the better. Hard to think of a better way to turn a user bitter than to humiliate them when they are helpless.
- the suspended user explicitly gives permission, e.g. to let the community challenge the suspension publicly
- the user lies about the facts of the suspension
I've also seen multiple times that users came to the MSE or to other sites to talk about the events, out of the limit of power of the mods. These posts were typically quickly downvoted and deleted, but I think no one wants this. In my opinion, the place where suspensions should be publicly talked upon, is the meta site where the suspension happened, and only if these criteria above hold.
My suggestion is: give to the suspended users a way to post a single question on the meta, while they are suspended. Once.
If they are sure that the suspension was unfair, they would have an option to challenge it - but, the price is that if they wouldn't say true facts, they could be easily proven by the mod tools. Furthermore, this post would remain a part of their profile forever (if it got a positively scored answer, which is likely).