It appears to me that in order to run your own Persona Identity Provider you must setup and maintain an SSL capable webserver for your email domain, equipped with a certificate that chains up to one in Mozilla's bundle (no self-signed cert), configured to handle the Persona protocol and authenticate you. FWIW, some (including myself) run email-only domains/servers with unnecessary services (httpd!) purposely disabled in order to reduce attack surface and administration chores.
AFAICT, even if you do setup your own Persona Identity Provider you would not have control over Relying Parties (websites you login to) and how they verify identity assertions. IOW, you couldn't prevent Relying Parties from taking the easy way out and issuing backend calls to Mozilla's verification service. Which would leak Email Address, Login Site, and time information to Mozilla. Nothing against Mozilla BTW, it's just a third party in such contexts and thus should not be privy to any information about account creations and/or logins.
I think those who run a strong browser config (limiting third party scripts, third party cookies, and/or cross site requests) would have to weaken their setup to even allow the Persona mechanisms to work correctly.