It would indeed make no sense to send all contents of all persistent containers per map while loading the map. It would waste a lot of network transfer time without any use for your client or player, and until you get to examine its content, it could have been changed already by anyone.
I am quite sure that the client only knows the content of a container you just even opened. I wonder if it is even notified when its content changes after you looked into it; you would probably have to open it again or try to change its content to see the new current content, except for items being processed while crafting.
Could be tested easily: Open a kitchen table and ask another player to put something it. Try the same with e.g. the crate at the merchant table near Harnquist's forge, which is no crafting container and won't process contained items. And the same with a furnace - who of you will be notified of the apple inside turning into ashes, only the one who put it in, or also the one who looked into it before the apple was put in?
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P.S.:
Just tested - everyone who looks into the same container is notified of updates immediately. As long as he looks in. I guess the server registers a callback when a client opens a container, and releases it when closing the view.