On-chain there basically isn't much (close to zero) privacy, but off-chain (LN's second layer) will allow you to enjoy more privacy than ever before.
I am quite interested how much it is going to affect the Bitcoin mixing services and altcoins which aim is to provide 100% anonymity. Monero seems to be the strongest one right now and I can't imagine people ditching it for the second layer solutions. It will take years for that to happen. There are still many people who believe in the full anonymity of Bitcoin on-chain transactions which is a common misunderstanding.