Personally, I haven't had any serious leaks that I know of so I've mostly suffered from increased spam and scam attempts (I know they were a result of a leak).

One time there was a leak from a university database and as a result there were a few news articles over the years about people that had their identity stolen likely due to that leak. It's not just credit card charges. They have had loans taken in their names, stuff bought on store credit or something (nowadays that's not so easy), stuff stolen from library in their name...

They had to deal with the fallout for years, always fearing that there's a new letter waiting at home regarding some unpaid expense or from debt enforcement agency that they have to contact and try to make it go away. It shouldn't be too hard if you have an open case with the police but it's not always that easy.

Also, if the leaked data is sensitive (e.g. private conversations, records about mental health etc.), you can face extortion or the data may get published.

One other thing that I know of personally is that victims of harassment very much don't like to have their contact info leaked to the harasser.