"Ownership does not always mean absolute dominion. The more an owner, for his advantage, opens up his property for use by the public in general, the more do his rights become circumscribed by the statutory and constitutional rights of those who use it." Marsh v. State of Ala. 66 S.Ct. 276, U.S. 1946.
While the Court has sometimes put public speech ahead of Private ownership rights, its by no means a decided issue for REA's purposes. The case above referred to a town that was completely owned by a mining company.
And since then, the balance has only really addressed such places as Shopping Malls in their capacities as "Modern Day Town Squares." So, the REA MAY be subject to the 1st Amendment as a private actor, however doubtful. The real issue is whether private facilities are subject to their respective state constitutions.
If you're super bored, here's a law review article that touches on the issue.
THE UNITED MALL OF AMERICA: FREE SPEECH, STATE CONSTITUTIONS, AND THE GROWING FORTRESS OF PRIVATE PROPERTY, 33 U. Mich. J.L. Reform 615