Blockchain.info keeps your wallet file which contain encrypted private key/seed, but they claim they don't have access to user's wallet and it's been proved with their open-source software even though i think there's no sure way to verify if blockchain.info use exactly same software within their website/app without add any backdoor.
As for other private info they store, i think only your email address, IP/browser details & mobile number (if you add one).
If you have privacy concern, i suggest you to use Desktop wallet, whether it's SPV or full nodes wallet such as Electrum and Bitcoin Core.
Seriously, I have read a lot of articles where many authors have claimed that blockchain don't really store the private key while many other authors claimed that it does. Its really quite confusing. Since they claim they don't really store users info, so I believed private keys will be part of it. Even if they do store it, I just hope it is well protected. But in the main time, I might consider moving to desktop wallet as you suggested cause I have been using web wallet since I started out in bitcoin and haven't received any bad news and I felt web wallet is less stressful for easy transaction.
As long as I know, the block chain doesn't carry your Private Key! I have been using blockchain.info since 2016, they only deliver unique Wallet ID then personally set 2FA security that's it.