Typically, we 'log in' to 'our accounts' on some company's server and use whatever fuctions the company grants us. They are not really our accounts in any meaningful way.
If one were to be banned by hackernew(or twitter), she would have no access to her data, social connections, et c.
In the urbit world, any data you have authored and any messages you have received are YOURS. They are forever available on your own server.
It is simply not possible for 3rd parties to ban/delete/control your conversation, when the model is peer to peer.
"How would one party prove they have the right data?" Before two urbit servers communicate, they exchange and verify public signing keys. Then, all messages are signed with verified public key.
If you mean what's stopping one party from changing their local copy of your data.... Nothing that I know of.
But they're the only ones who will see their local data.
It's not worth it to me, I'd rather do other things and this is almost absurd levels of exclusion but I like the obscurity-as-a-quality-control method approach in general for some projects.
The whole Urbit system could be described in terms of linked-lists and S-Expressions. Even languages as obscure as K, J or APL have better reasoning than Urbit.
Obscurity only leads to smaller number of people wanting to explore something. Only people who stays with it are "believers" and not experts.