So you open 2 channels on different devices: 1 channel to Bob, who runs a well-connected node, and another channel to Alice, who also runs a well-connected node. From your device connected to Bob, you send funds to your device connected to Alice. After you close the channels, there is no direct blockchain connection between your transactions.
But you can't receive on your channel to Alice until you make some payments, and you'll be able to only receive as much as you pay. So you basically get back at spending your bitcoins on the LN.This also assumes the output you used to fund the channel to Alice is not tainted in any way.
So the question is: who's going to keep track of all LN-transactions, and for how long?
What do you mean by LN-transactions ?If you mean LN payments, they (normally) cannot be linked.
If you mean Bitcoin transactions opening or closing (in any way) channels, then I think already-mentioned-above companies already do.
And although I'm not sure if LN works like this: if your channel to and from the node is not public, only Bob and Alice know the exact connection.
You should not rely on unpublished channels, they are biggest (potential) privacy leaks than they seem to be (in addition to having many other downsides). (Non-related to the initial subject, but if you are interested 
https://github.com/lightningnetwork/lightning-rfc/pull/681) 
But privacy features *are needed* a step before accessing the LN anyway, and it is not an alternative in any way.
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