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.
The platform has responded to the EU's anti-money laundering pressure "Especially the Finnish Parliament," although details of the verification have not been made clear, the platform will hurt to sacrifice some privacy.
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.
It's not a particularly surprising thing that they are doing.
In the past, they've already tightened up on KYC/AML in numerous occasions by first asking for KYC for all advertisement listings, then asking for KYC for traders with significant volume. This new policy that they may be implementing right now is simply a confirmation of the regulatory pressures that are put on them, and an extension to previous policies.
Whether they're going to go for the home run and just ask everyone to flat out identify themselves, I don't know. Perhaps it's more likely they'll just ask more personal information without necessarily requiring proof of them when you sign up, up to a certain threshold. But this new policy may well lead to more people jumping ship and looking for alternatives, but the fate of each platform will be similar.