In Bartnicki v. Vopper, SCOTUS held that a radio station had a 1A right to distribute stolen property--an illegally wiretapped conversation. Because the recording came to the station over the transom, the station was not involved in the illegal wiretapping itself, just like the Ashley Madison mirror sites today.
On the other hand, the Bartnicki court recognized that there were "important interests to be considered on both sides" and pointedly did not consider "domestic gossip or other information of purely private concern." In that case the illegally intercepted conversation was of public concern.
If this ever goes to SCOTUS (unlikely) I suspect it will favor the 1A arguments because of the unique nature of Ashley Madison. It's not a dating site for singles, but a service for "discreet married dating," after all, which necessarily implicates the other person in a marriage. And the current SCOTUS is probably more 1A-protective than it was in 2001.
Also all the allegations lodged against the mirror sites could be lodged against news organizations that are reporting on the data dump and naming names. So the plaintiffs will have to persuade courts to invent a new theory of liability that sweeps in mirror sites but not news organizations, a non-trivial task. How about a news organization that publishes hundreds of names?