I doubt anyone would buy fannie or freddie nor would any banks seriously compete with them.
Their re-insurance business is very high risk, will require huge capital reserves and provides a low rate of return. Once fannie and freddie go through the pain of raising enough capital to be released they will do OK, but banks won't be willing to set aside that much capital for the small fees fannie and freddie earn.
I think they will continue to have this business to themselves, unless they decide to get out of the business too.
The "brand name" of fannie and freddie isn't really worth anything. Their customers are mortgage originators who want to sell off loans to free up capital to write new ones. They are well aware of fannie and freddie, and would be well aware of any newcomers competing with them. Normal home buyers might have to meet "fannie mae" loan requirements from their actual lender, but other than that annoyance would have nothing to do with these re-insurers.