In this case, "property rights", as you put it are at odds with free speech. I don't think it would be ideologically inconsistent to believe that free speech trumps property rights in cases where the two principles conflict.
I also think there's a lot more nuance to this situation than you seem to be implying. I very much doubt there are a significant number of people ("right wingers" or otherwise) who believe property rights are the most important concern in all situations, nor are there many who believe in an absolute right to Free Speech at any cost. (The constitution itself allows for narrow exceptions for both of those rights.)
