Well, as I understand it, the RIAA needs to be able to prove that it's more likely than not that the download in question was initiated by a particular user, so on a public wi-fi network, there's a bit of a proof problem unless the RIAA is able to get a hold of the actual MAC address of the computer that's downloading (and/or uploading) the copyrighted content and then to connect that MAC address with a particular computer. I am not well-versed in computer forensics though, so don't really have the knowledge to tell exactly how they do that.
At the same time, the operators of the network connection points over which the infringing material passes (and this includes public WiFi operators) would not be considered to be infringing, and would bear no liability solely for providing the network connections through which infringement took place. For some of the reasoning behind this, see Religious Technology Center v. Netcom, 907 F. Supp. 1361 (N.D. Cal., 1995).
Could the public Wi-Fi operator control access? I think that the answer in most states would be 'yes'. Privacy laws in the US aren't at all very restrictive of a public Wi-Fi operator's right to monitor or restrict internet traffic through an access point which they own.
And the method by which RIAA monitors internet traffic (through the P2P networks, at least), is to search for and log IP addresses of computers which are hosting infringing content. Since those IP addresses are already out there, and publicly-visible (for the functioning of many P2P networks, those IP addresses need to be publicly-visible), there are no real privacy concerns. Once those IP addresses are obtained, the RIAA generally tracks them back to the ISPs to which the IP addresses have been assigned and then subpoena the user information from the ISPs. While many ISPs do have legal agreements which obligate them to protect consumer privacy, the courts do have the power to require them to surrender their user logs, even if it would otherwise be a violation of the privacy agreement. Consequently, many ISPs actually state in their privacy agreement that they will not be held liable for cooperating with lawful court orders (although they probably wouldn't be liable anyhow, just due to public policy, it always helps to have a contract the user agreed to that can be pointed to.)
Should the ISP move to quash the subpoena, the issue of user privacy might come into play, but as far as I know, the court generally sides with the RIAA's right to ferret out people who are engaged in violations of the law. I suspect that a subpoena of records from YouTube, in which the RIAA, MPAA, or other body was requesting the identities of actual unknown defendants (like in the P2P cases) would turn out much the same way.
And certainly, there are privacy concerns surrounding these issues, and there are groups (such as the EFF) who are agitating for stronger privacy protections and measures for Internet users, but there's not really all that many legal stumbling blocks in the way of a large industry body (such as the RIAA) getting the information it needs to pursue direct copyright infringers.