Citizenship may matter less than you think. Many post-docs expect to get something back from you for the funding you get. If you can do the required work, it doesn't matter much where you are from.
There may be some, of course, that favor citizens and some (national defense work) that require it. And getting appropriate visas has become a problem for many.
But, don't assume that you are ineligible without at least asking or doing a bit of research into positions that are offered. I'd expect that getting your own funding is actually harder, unless you have support from your own government.
I think that self funded post docs are pretty rare and, perhaps granted sometimes as a courtesy to someone who needs an academic home for a while and can, perhaps, assist some regular faculty in their research. But, again, funding strictly out of the candidates personal funds would be more rare still, but with some people bringing funding with them from another institution (or government). It could be the only option for some who don't qualify for a "visiting professor" position for some reason. And few institutions are likely to have a regular process to cover such a case.