With a lot of effort, tbh. And that's just to see it happening - making anyone
care is orders of magnitude harder again. People have enough to be worrying about these days without online forum drama.
To answer your question a bit though:
* The 'dead giveaway' is basically always going to be some kind of mod drama. Sometimes this is fake drama, sometimes it's to harass existing mods out, sometimes it goes as far as print media and tv stories accusing a forum of bad behavior. Look at who is instigating it, how often they support each other, how much they're posting and commenting, how quickly they comment on relevant threads etc.
* Keep an eye out for sudden and extreme rule changes, such as moving political talk to a seperate (much smaller) subreddit moderated by known antagonists; or removing discussion of topics such as racism, BLM, etc. This can be a sign that the new mods are 'bought' in some way.
* Browse by new to get a glimpse at stories before they're removed. This is most useful when there's a story getting actively buried, or a narrative being pushed.
* There are tools that can help you keep track of users such as RES, or to analyze user's history.
* You can use these tools to find people who are fighting back and keep track of their insights, or keep an eye on accounts known for bs-ing. It's a lot of effort for very little reward though lol.
* It can be illuminating to post verboten stories yourself, then keep track of any strangeness. E.g., the story getting removed, funniness with the vote count, particular commenters rushing in to spread fud, etc.
You can try it here on HN, to get a glimpse of how actively some groups keep track of hot topics: Find a fresh story that goes against an industry (or empire) known for their online PR goons, and see what happens when you post it. Even comments mentioning certain keywords will attract attention. If you think that sounds conspiratorial - just try it.