If I was making a map and wanted to test what I had with multiple players to make sure it works, would I have to publish it first, or can you host custom maps without publishing them?

"Private" means it simply is not able to be "published" (in the lobby, make game public button). Still, when hosting private, you can invite friends and you can test the map with em.
Before releasing the game while being in the lobby make sure you chose the right game variant! You cannot change it afterwards.

I'm a bit slow. Does this mean we do have to "publish" our maps to test them, but we can publish them as "private?"

You don't have to publish them to test them, you can do that in the editor by hitting the SC button at top right. To test them online, yes, you must publish them, and you can publish them as private if you want to.