Privacy
What is private?
Payment transactions within the Lightning Network are private as they do not reside on the Layer-1 blockchain (e.g. Bitcoin blockchain). Information about payments is kept in the Lightning nodes in a private fashion. The individual payments between two parties are hence private.
Depending on the routing implemented and used, payment channels can be public or private, but by default payment channels are usually public and visiable to any observer.
What is not private, what is public?
On the other hand, the creation of the payment channel exposes the funding amount (how much), the funding address (e.g. public BTC address) and the date/time on the public Layer-1 blockchain. Similarly, the closing of channels exposes information such as the settled amount (how much), the receiving address (e.g. public BTC address) and the date/time on the public Layer-1 blockchain. For more see On-chain transactions.
In Lightning Rendezvous Routing exist private channels which are special payment channels that are encrypted.