Localbitcoins does escrows so it logs the transaction details, so it is not anonymous.

These records must be kept for a long time and will be given up if ever requested by law enforcement or tax authorities. They could even eventually be gagged and not allowed to discuss giving up their records so we wouldn't even know it happened.

That is why I recommend TorATM. Full disclosure, I am apart of the staff of this website but it was built for this exact reason. We wanted to increase anonymous access to bitcoins and allow people without bank accounts to obtain bitcoins. We delete all records after a short waiting period and purge identifiable details immediately.  Our identities are not public knowledge so we cannot be easily subpoenaed.

 http://toratm3dhalotm3f.onion/guide