One Reddit user has looked at the issue and discovered that these private worlds may actually just be existing servers which then make your existing session private.
“I went to grab a bunch of clipboards in my private server, I disconnected and came back in. Clipboards were back.
I then have just this minute gone to a PA spawn, it was there, I didn't pick it up, I have one of every set I don't need it. I hopped 3 times and got a server it had been looted and wasn't there.”
Additional investigation must be done of course, but this isn’t a good start to Bethesda’s subscription. While the official blog post doesn’t outright state that your private world is indeed a new server, the expectation that your subscription goes toward a new server would not be unfounded, and could be viewed as misleading from Bethesda.
Additionally, this is corroborated by other posts such as this unofficial bug list,
"Issue: Multiple Reports of "New" worlds created are not New. (Ticket Submitted on 10/23/2019)
There are a few reports and I've got confirmation from a few reputable sources that "newly created" worlds when a Private world is first selected by a "Fallout 1st" player has Dead NPC's, already looted Junk (not the timed spawns but Clipboards) and other abnormalities that when tested multiple times seem to indicate these are not "New" world instances but re-allocated older Server instances."
Further below, it states that these servers aren't even actually private,
"Issue: Private Servers are in no way Private. (Ticket Submitted on 10/23/2019)
Without the "Anonymous" mode we've begged for, and the fact somewhere along the way "Temporarily Blocking" someone on our friends list no longer blocks them from seeing us anyone on the Friends list of a "Fallout 1st" player can join.
This is problematic for those with large friends lists who are trying to do a small organized event, "Merchants" who just want a couple hours of not being pestered by everyone who wants a trade so they can play the game, etc."
It has been a rough start to Fallout 1st, with news of the unlimited scrapbox deleting player inventory just this morning.