I'm a more casual user, not as experienced as others on this thread. But since I'm an old fogie with a POP account, maybe I can contribute.
So I do use POP email and Apple Mail. Everything is on my one computer (and its backups).
I can read and retrieve all my previously read/retrieved e-mail even if not online, so that lends credence to the "once it's on your Mac it's on your Mac" theory.
There are times I'm away from Internet for a few weeks, and I can still look through and read all my old e-mails (dating only back to 1999, so I'm a pup compared to
@Fishrrman 
)
This does explain something: I never understood why there even was an "On my Mac" set of boxes in my Apple Mail program. Like why? Because
all of my e-mail (even the inbox) is "on my Mac." Now I finally understand that (because that doesn't happen if you use something other than POP mail).
And yes, I can tweak a setting with my ISP that says how long they will save new mail on their servers after I have retrieved it. When I was moving to a new computer or having some sort of difficulty I would set that to a longer time period, and then I could "re-retrieve" those e-mail in case I needed to. Mind you, that doesn't help with anything in your "sent" box (at least I don't think so).