https://www.vice.com/en_us/article/qjbagd/eu-lawmakers-back-rules-to-identify-owners-of-crypto-wallets

On Thursday, members of two European Parliament committees voted to approve draft legislation that could require exchanges to verify the identities of the owners of unhosted wallets―wallets that aren’t controlled by exchanges or other custodial services, and which the owner has sole control of the private keys. 

According to a press release, politicians on the Committee on Economic and Monetary Affairs (ECON0 as well as the Committee on Civil Liberties (LIBE) voted 93 to 14 with 14 abstentions to approve the proposed rules for crypto. The rules not only task firms with identifying the owners of unhosted wallets, but also asks them to report any transactions to anti-money laundering authorities with no minimum amount, meaning that the rules would apply no matter how small the amount being transferred is. The rules would not apply to "person-to-person transfers conducted without a provider," the release states, "or among providers acting on their own behalf."