LN is an onion routed network, and has some privacy - notably, sender privacy, as the sender does not need to be known to the recipient. The recipient however is not private, but there have been proposals for rendevouz routing which could also anonymize them.

LN payments are based on Sphinx[1], but they don't leverage the full capabilities of it. Sphinx only allows for single-use replies.

The proposals for bidirectional anonymity involve using HORNET[2], which builds on Sphinx (specifically, it uses Sphinx's single-reply message to establish a HORNET connection), and enables sender-receiver anonymity and bidirectional transfer beyond single-reply messages.

It has also been proposed to include the use of TARANET[3] to prevent deanymization via traffic analysis.

[1]:https://cypherpunks.ca/~iang/pubs/Sphinx_Oakland09.pdf

[2]:https://arxiv.org/pdf/1507.05724v1

[3]:https://ar5iv.labs.arxiv.org/html/1802.08415