Hate Speech is protected speech under the First Amendment to the US Constitution according to the Supreme Court, so if someone posts a racist slur in a forum hosted in the United States, does the First Amendment protect that speech, even if the forum is run in another country....but again, hosted in the United States?
The First Amendment is not as 'open' or liberal as most people think it is. https://www.law.cornell.edu/wex/first_amendment
There have already been incidents of people in countries like the UK being prosecuted for making comments on US-based sites like Facebook. What you do have to realize about "hosting" is that can be a little bit of a blurry concept here. A site might be "hosted" in one country, but interface with that site is also going to be "hosted" on internet servers in other countries (otherwise people in other countries often would not be able to access it, or the connection might be extremely slow). The First Amendment in the US would not do anything to protect you in other countries. One of the huge issues about the concept of "hate speech" laws are there could potentially be some very concerning jurisdictional issues. So far there have not been any instances of that, but there almost certainly inevitably will be in the future. I personally believe that it's a huge huge slippery slope, and will lead to an erosion of further liberties in the future.
I don't think it really matters where it is hosted, the First Amendment protects free speech of American citizens or residents from restriction by US federal or state government regardless. It's also always worth remembering that free speech has never been unconditional and there is long established precedent for legitimate restrictions in specific circumstances.
It depends on whether the owner of the forum tolerates hate speech. If this is the case and the server is in the USA, those affected from other countries have little chance of preventing this in court. Does anyone know the German Thiazi board? A neo-Nazi forum in the USA, around 20 years ago. It could only be prosecuted because copyright-protected music was illegally distributed there. All the Nazi ****, hatred of Jews, and Holocaust denial didn't interest the US authorities.
