A lot of private organizations, websites etc. get around the first amendment by saying that since they are a private organization, therefore they can decide what you are allowed or not allowed to say.
While that is certainly the law, there are cases in which private institutions should protect free speech even if they are not required by law to do so.
But first, let’s bring up an example in which a private venue doesn’t need to tolerate free speech: eg. a video game. A video game about flight simulators.
The purpose of flight simulators is training as well as having fun. When you are playing a flight simulator, 99% of the players don’t want to hear about any political or religious nonsense, so, in order to ensure everybody has a good time flying, the community can choose to ban people who bring up controversial topics.
This is true for most recreational venues where people just want to have fun. Restaurants. KFC doesn’t have to tolerate my racist bullshit, blacks need to eat their chickens in peace!
BUT, if your organization is academic or a forum about free discussion (especially WHEN YOU TOUT FREE THINKING), then you have to be very careful about free speech. A debate venue is not like a yacht club. People go to schools, to discussion sites etc to learn, to open their minds, and there, ALL RELEVANT SPEECH, yes, EVEN “HATE SPEECH”, should be available fully uncensored.
If your organization is called “skeptics” or “free thought”, then Creating a comfortable environment simply shouldn’t be your first priority. I talk a lot about the marketplace of ideas, but a forum literally is a marketplace of ideas. In the marketplace of ideas, ideas stand on their own merit. The better it is, the more people will adopt it. The worse it is, the more it will become obsolete. If there is an alternate idea, both ideas compete until one wins.
Should private institutions protect free speech? Depends heavily on the type of private institution. But if your organization is about ideas, controversy, etc then generally speaking, YES.