Without getting too in the weeds on this (i.e. without covering every nuance, identifying every exception, and walking through how the law has evolved over time on this issue), here's basically how it works: The Supreme Court has said that First Amendment rights (i.e. to free expression) don't apply in the context of private property, even when that property is used to serve the public or has been opened to the public generally. To paraphrase the Court, the Constitution does not protect against private parties (to include corporations) that might abridge the free expression of others. But to clarify, I'd make a few additional points:
(1) That does not mean that laws can't be implemented (e.g. by states) that to some extent protect people's free expression rights against certain private parties. Such laws (or for states, state constitutional provisions) can raise balancing issues as between property rights (e.g. based on the Fifth Amendment) and speech rights, but the Supreme Court has said that they can in some circumstances be implemented. I don't think they are all that common though.
(2) We should distinguish between private property being used by the public and private property that provides a public function. Based on the so-called public function test, the Supreme Court has concluded that in some quite limited contexts private parties are functioning as the government, and in those contexts constitutional free expression rights do apply. For instance, and I don't know if these situations even exist anymore, when it comes to company-owned towns this is true. Those would be where entire towns - residences, streets, all the stores, all the infrastructure - are owned by a single private corporation. The Court found (a long time ago) that in such cases the private parties were functioning as government and providing government services just like they were public municipalities. That same principle does not apply, e.g., to privately-owned retail stores though. I don't even think it applies to, e.g., large shopping malls anymore though at an earlier time the Court was starting to expand that exception to the general rule to cover a broader range of circumstances. More recently the Court has repudiated that line of reasoning and narrowed the exception back such that it only includes company owned towns.
(3) When private parties take public money, that can also affect the consideration - e.g., such as the case with universities.
(4) Just to reiterate, public accommodation is not the same as public function. With regard to the former, it does not subject private property to constitutional restrictions. Those limitations apply against governments and entities acting as government. Laws can be passed that extend what we think of as constitutional protections to public accommodations, but generally speaking there aren't such laws at the federal level when it comes to free expression rights. Private parties generally can, e.g., prohibit people from trespassing on their private property based on speech. The same is true when it comes to Second Amendment carry rights - at least until new laws are passed preventing it.