Localbitcoins is one of the oldest markets to buy/sell BTC, and perhaps one of the last places where you can get your coins anonymously.
Anonymously? I don't think so. Most of the payment methods on LBC are Bank/Wire Transfers. I doubt there is any anonymity there when you're literally sending your bank details to the seller/buyer.
As announced earlier, LocalBitcoins is working on improvement measures that will allow us to provide a safer and better service conforming to the regulations. On Monday 18th March, we have launched a new account registration process where users can verify basic information already during sign-up, making it easier for the newcomer to find trading partners from day 1 and increasing the number of suitable customers to advertisers as well as inhibiting the creation of illegitimate accounts.
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That is good indeed! Way better than asking for KYC's when a high volume withdrawal is requested like other exchanges.