Online login services are generally many-to-one. For example, many sites accept Facebook login, but for a user to log in that way, there is only one identity provider they can use: Facebook. If you don't have a Facebook account, or don't want the site in question to have any access to it, you can't use Facebook login.

When the article says "distributed", it means Persona is many-to-many. Any domain can implement the protocol, so when a site accepts Persona login, you can choose from many identity providers – including your own, if you're industrious and want to set one up for your domain. Most people are using Mozilla's service today, but the idea is that email providers like gmail will implement it themselves in the future.

It's used in the sense of "decentralized": no one company controls the Persona system as a whole. Authentication is performed against the domain of the email address the user gives, falling back to authenticating with Mozilla's central server if and only if the domain doesn't support Persona yet.

Pretty sure it means:

It doesn't have to go through Mozilla's servers; anyone can set up their own instance and work with the other ones.

No single authority is needed to vouch for a given identity, instead numerous authorities can validate an identity. That's my understanding.

Distributed identity seems to be notoriously tricky to fully understand for both end-users and developers. I still don't really get OpenID.

I guess they mean that you do not have to use Mozilla servers and can setup your own one (with a retina scan...). I still prefer using theirs as it is less hassle and the same identity can be used across multiple websites.

Not locked to a central service provider.