It doesn't have to be the gateway operator. Maybe TTN operates a central registration of devices, then they can distribute a whitelist (or the gateway can look up a DEVID and see if it's known to the network), and pass packets if it is. This way the gateway operator doesn't need to know the owner, only that that DEVID is valid/known.
As posted earlier, SIGFOX issue a PAC (or whatever it's called, which is a per device activation ID), which can ONLY be issued by the operator (in this case SIGFOX or the local country operator which in the UK is Arqiva). Once you have a device and the PAC you can register it on the back-end and utilise the network. If you then say give your device to another user, it won't work as the PAC is assigned to you and is non-transferrable (i.e. it's tied to your back-end account).
So TTN could register users and DEVID's associated with them, then the gateway would say query TTN when a DEVID tries to send data through a gateway, if it's registered, fine it allows data through, if not, it drops them. Obviously a user would need the ability to de-allocate a DEVID too.
Then if something illegal was done, the gateway operator would just point the police/whoever saying it was a valid DEVID and they'd contact TTN to get user information.
This doesn't of course cover using a gateway on your ISP connection, which may not be allowed, but different kettle of fish and it does mean there no anonymous access.