VLS itself doesn’t deal with privacy, so like Tony noted this would depend on the privacy practices of the LSP or custodial app in question.

There are other setups, such as an enterprise running their own node(s), where they’d want to separate the signer for security. This provides them security both from malicious actors attacking the node but also internal employees, to allow for separation of duties between node operator and the person who can sign for transactions (or people once multi-sig lightning becomes a thing ⚡️)

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.