How do ESI and public/private cache work together?

With ESI you can decompose an entire page and treat the parts differently.

LiteSpeed Web Server allows you to store content in either a public cache or a private cache.

Combine these two elements and you get something very powerful. You get a system that can split a page into public and private parts, cache each part accordingly, and then reassemble the entire page's content from the relevant caches and serve it to a user without ever hitting the PHP backend.