"No, no, we don't want to silence you; we just want to remove the hearing ability of everyone around you until you stop talking."
There's an argument to be made that even privately owned entities are to some degree still stewards of the public trust and therefore owe some duty to the public, especially if they benefit directly or indirectly from public subsidy, and/or if they are able to exert monopoly power. This is generally expressed through regulation.
There's another (albeit related) argument to be made that if a platform for speech[1] is provided to the public, that it should conform to the same standards of freedom as elsewhere, including government. I believe this is what is mainly being debated here. Attempting to comply with the spirit of such an argument can be challenging for a company whose platform spans multiple "elsewheres", but there is also an argument in favor of greater freedom of speech, no matter the local standard (which I personally happen to support).
[1] or, perhaps, anything, such as a space for gathering
Many people seem to often not notice how never before in human history has centralised communication existed, and it was never foreseen that a few "private" companies would control almost all of it.
It's why I always think of it as rather rediculous when people refer to the U.S. constitution and other literature and preach "but X says that only a government is controlled and private-company-hosted speech is defenseless!", and not realise that Facebook et al, if anything, _is_ a "government" to a 17th Century human (especially to those who drafted the Constitution).
If power is a measureable "quantity", then I'de say that FB alone has more of it than what 17th, 18th, and 19th Century Governments had. The laws are very outdated I think.