its a multi layered process. mostly explained in my first post.. but ill emphasise some stuff
BTC does not leave the bitcoin network
its locked.
people on LN make payments in different formats/units of measure(tokens) that are not bitcoin formats/units.
what is done is when a LN payment is successful they then both contribute a signature for a bitcoin(segwit multisig) formatted transaction they call a 'commitment' which is a TRANSLATION of the successful msat payment.
which they choose not to broadcast.
and instead make new payments and reveal the private key(revoke key) of that last commitment so that if someone did try to broadcast to the bitcoin network yet was not the correct or most updated commitment.. .. IF it got confirmed. its still on a lock.. and the victim could IF they were aware, awake.online or using a watchtower service. could use the revoke key(private key to the spend) to spend before the lock time expires back to themself. thus not letting the scum thief take the funds not deserved to them (emphasis on all the IF's)
however this does not make LN private.
nodes on the LN route network are public and are given nicknames to identify them. so people can map out routes between channels/nodes..
when a channel is created or when a payment is successful they update the gossip protocol to tell the network there is something new about the channels.
there are many flaws of LN that allow people to reveal the 'gossip' of the channel updates and announcements and all the routes available to see when certain routes all send updates to the gossip protocol in quick succession. this can reveal when certain routes are used.
also by 'faking payments' along certain routes with differing amounts and cancelling it and repeating it (brute forcing) you can find out what liquidity each route has and how it changes over time. and piece together alot of info about where liquidity moves to and where it stops. so you can form a list of payments that have been made by others
alot of 'chain analysis' companies are already making algo's to do just that to form a map of nodes and channels and balance and watching where balance changes occur to then list which person paid which person and how much.