Images may not be copied or offloaded, and the images and their texts may not be published. All digital images of the Oxyrhynchus Papyri are © Imaging Papyri Project, University of Oxford. The papyri themselves are owned by the Egypt Exploration Society, London. All rights reserved.
Future use of the transcriptions may be hinted at a bit on the About page:
The papyri belong to the Egypt Exploration Society and their texts will eventually be published and numbered in Society's Greco-Roman Memoirs series in the volumes entitled The Oxyrhynchus Papyri.
It should be noted that the closed nature of the project is likely a side-effect of UK copyright law, not a policy decision by the Zooniverse team. In the US, a scan or transcription of a public domain work is also public domain and not subject to copyright. In the UK, however, scanning an image creates a copyright in the scan, so the up-stream providers automatically are able to restrict down-stream use of public domain materials. In the case of federated digitization projects this can create a situation like that of the Old Bailey Online, where different pieces of a seemingly-seamless digital database are owned by entirely different institutions.
I will be very interested to see how the Ancient Lives project fares compared to GalaxyZoo's other successes. If the transcriptions are posted and accessible on their own site, users may not care about the legal ownership of the results of their labor. They've already had 100,000 characters transcribed, so perhaps these concerns are irrelevant for most volunteers.