I can think of several Kdramas that fit that, I can think of dozens. But you didn't ask for Kdramas.
Can't think of any Japanese dramas. They are usually about ordinary people, or murder mysteries, or both. Or have an insanely implausible premise. The press doesn't seem to have much presence in the story telling, and when it does, it often reflects on the characters in a favourable light, or is perhaps used for comedic effect. If a character's reputation is ruined, it's through word of mouth or rumours. Or they deserve it, but that's another kettle of fish. This appears to hold out in other Japanese media forms - including anime, manga or any of the light novels I've read, which all together, number in the hundreds. I could be wrong though, there's a LOT out there I'm unfamiliar with. And my memory isn't perfect, it may have simply not left a strong enough impression.
(Though there's probably a few if you want the characters to deserve to get their reputations smeared. They'll be anti-heroes, and will definitely have a sad ending... Likewise, a character who has a false reputation, but not necessarily targeted by the press may be manageable.)
You'll probably have more luck with Cdramas. I can't name any, I'm not particularly interested in them. I've read several synopsis that sound like they may be what you're looking for, but I never remembered their titles; I wasn't planning to watch them.
