Where I live (Poland), only the person itself can request their criminal record from the state. This is a routine procedure required by some employers, you can even do it online these days.

Most if not all criminal offenses "expire" after some years, how long depends on the offense. If there's something you've been charged with but not convicted of, it doesn't appear on the record.

This is easier to implement for us because there are limitations on how media can report on criminals (no last names for example). Even in the US, I think that system could be workable. Instead of attacking distributions of "unedited" criminal records, you'd have to target employers and require them to only acquire the state-approved versions.