LSP is starting to become a loaded term. Depends on the service the LSP is providing. Is it hosting, is it just liquidity, etc.
If it's just for liquidity / channel management, then they see the balances of the channels you have with them (some wallets like ours let's you connect to anyone, but not all have this). They'd also see when you are making a payment or receiving a payment but they won't know where to or where from.
Hosting based LSPs like greenlight probably have a way different privacy trade off that I have not deeply explored yet but I'm assuming they see everything (besides the sender of an incoming payment). Custodial wallets would be the same way.
Except custodial wallets probably know some sort of identifier that can be associated with a real user. Most non-custodial lightning wallets that I know of have pretty much no email requirements, associations, etc. besides the pubkey of the node.
Hosting based LSPs like greenlight probably have a way different privacy trade off that I have not deeply explored yet but I'm assuming they see everything.
This is what I assume too. If true, VLS via a greenlight-like service is amazingly non-custodial but less private than using a noncustodial node with channels to just an LSP because the node provider knows where payments are going ... but maybe this is where a 3rd party trampoline could be used.
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