They are private in that they are not run by the government and not subject to the First Amendment right of access that permits basically anyone to walk into a courtroom for basically any proceeding and review virtually all records of the proceeding.
They are not by law private in the way you're talking about them -- some sort of legal prohibition on disclosure of the proceedings. Such a law would probably violate your First Amendment rights.
But you are permitted to contract away your First Amendment rights, and practically speaking, arbitration may force you to do that, especially if your dispute is solved through an arbitration-imposed mediation. Most of the major arbitration associations include confidentiality clauses prohibiting the mediator and the parties from disclosing what happens in that process. For example:
If the dispute skips or leaves mediation and ends up in an actual arbitration, those rules don't typically apply, but the party would still be subject to the arbitrator's orders for conducting the hearing, which often include a broad gag order preventing the parties from discussing or disclosing what happened in the proceedings.
So the answer to your question, as always, is that it depends. Is there a confidentiality requirement in the rules of the arbitration association, in the arbitration agreement, or issued by the arbitrator?
If the answer is yes, you'd probably need to get permission from the arbitrator, the other party, or the courts to disclose what happened.
If the answer is no, you should probably look harder for the confidentiality agreement that you probably missed. If the answer is still no, you should probably ask a lawyer to double-check. If he says no, you're probably good to go.