The online component of Persona only matters in the interim (nothing it provides is necessary to the protocol, it is all about lowering the bar for implementers/providing support for browsers without a built in implementation). In the long term, you have to worry about whatever identity provider you are using.
Nothing depends on Persona. For example, see https://persowna.net/, which I wrote and which you can use with your own domain for authentication. You can also install your own ID provider on your site and not rely on any third party.
