From my interpretation, in order to get the data you 'own' (it's not theirs to give away since they can't claim the copyright on it), you need to use their services. The agreement the user has with Antropic is for the service, not a restriction on how the data is used.

not being able to train in it is a restriction on using the data though.

if i publish that data, and someone else trains on it, am i liable? am i responsible to ensure that noone trains on that? how am i supposed to enforce that?

so if i publish that data, and someone else trains on it, am i liable? am i responsible to ensure that noone trains on that? how am i supposed to enforce that?