there is no absolute safe place. I also ask this question for their admin, they reply to me:
"Just start a dispute, and it is easy for us to understand who is providing fake evidence and who is telling the truth."
another word, your safe is in localbitcoins' hands.
Ye well a dispute is useless if i understood because they won't refund you so im not sure what is it for (probably the other party getting banned aswell but that's not getting my money back)