If lightning network transactions are "off chain", can we still see the transaction history then? If the lightning network is a off chain system. Are the transactions private or public? And does that mean once you send bitcoin to a lightning network address, it can possible get hidden in a payment channel? Or is that not possible?
Are the transactions private or public?


Mostly private. If you are not involved in a transaction (i.e. not a sender, receiver, or intermediary), you really won't know anything about the transaction. Transactions occur off chain so there is no permanent record of a transaction happening. All that outside observers will see are the channel opening and closing transactions.


  it can possible get hidden in a payment channel?


Yes, you can have a payment channel which no one else knows about, at least until it closes.. These could even exist today and it would not be obvious until the channel closes. And if a channel is closed cooperatively, it will look just like any other 2-of-2 multisig transaction. So even then it is not clear whether a transaction is a closing transaction for a payment channel.