I recall a few finance papers saying (paraphrasing) "approximately all private info is reflected in stock prices". The same is obviously true of prediction markets. If government officials are banned, they'll just give a little wink to a relative or friend, and the prediction markets will reflect the private information.
On a personal note, I find these markets incredibly useful in day to day life. There's been a joke for a while that putting site:reddit.com in your google search is the only way to get (some) real information. It's becoming true that putting <search terms> AND (kalshi OR polymarket) is how to get accurate info.
I think you're thinking of the efficient-market hypothesis. The hypothesis is that prices reflect all available information, and "available information" is the key part. The "strong form" includes private information, but research has not found support for this. And even the "semi-strong form" falls apart. For instance, the market for small cap stocks is not as efficient as the market for large cap stocks.
You need a market that has enough people paying attention and doing the work, and you also need a market that has enough liquidity.
Source: I have a PhD in capital markets.
Accurate info on what? Most markets have almost no liquidity, outside of sports and crypto
Political outcomes is the biggest. Businesses (and individuals) can be strongly (financially) affected by who's elected. I'd been very curious in the past about political outcomes but had never used prediction markets, so I built a little tracker that inputed the probabilities based on scraping ~14 betting sites. I had to do it on a market-by-market basis, and there were lots of edgecases. Having them all in one place (or two; kalshi and polymarket) makes life much simpler.
Prediction markets have also directly affected my travel planning, helping me avoid a country that wasn't at war at the time but had (in my judgement) a too-high risk of war.
You're completely right about thinly traded markets; those are way, way less accurate. But they also offer the greatest opportunity for someone to bet the other way, which has the effect of 'correcting' the market.
Elections usually also have good liquidity.
https://en.wikipedia.org/wiki/Efficient-market_hypothesis