Where did you come up with the idea that free speech laws give them the right to censor. It doesn't. Ownership and private property rights give them the right to censor.
Whether liability and damages are assigned would depend on the details, but the principle is quite clear here.
Otherwise yes, it is quite possible to be liable for whatever happens on your private property. If drug deals are taking place between third parties in your house won’t you be liable? If you allowed those parties there you most definitely will be.