Anonymity from who is the question.

With a custodial option they see everyone you pay to, as well as every receive you get.

Doing it with your own node today will share your node ID with anyone trying to pay you, but outside of that it takes a much more sophisticated attacker to see sends and receive origins.

There are steps you can take to mitigate the on chain info shared when using your own node, but it's certainly not a panacea right now.