Obviously I don’t think the answer is to just wait until a listing for a specific film shows up on eBay
Actually, yeah, most of the time that’s the answer.
Is it some place like private collections
Yeah, some of the prints that have been scanned were in the hands of private collectors, and there are even some private collectors who will allow one of the members of the restoration teams to view the prints but not scan them, as it’s not technically legal to own theatrical film prints of copyrighted movies. They don’t want Disney or Lucasfilm or Fox (heh…guess that’s all the same thing now) to catch wind that they’ve got it and raid their collection.
Then there are some who will allow their prints to be scanned, but not released until specific identifying characteristics of that particular print are eliminated.
But no, most of the time it’s eBay or the like, and print scanners rely on donations from people like us to acquire them and pay for the scans, since it’s not cheap.
and local archives that allow you to rent stuff out
Nope, not that I’m aware of. I don’t think theatrical film prints can be rented from any sort of archive, at least not for this purpose.
And where would I even start to look for these things in my area (or further beyond, for that matter)?
I think the above answers have mostly covered that…well…you don’t. You can scour eBay for prints to buy, and you might be able to get in with collector communities online once you have some prints in your collection, but it’s all rather underground due to the less-than-legal nature of the hobby, so I don’t know where you’d go to find said communities.
Hope some of that was helpful, and that if any of that is inaccurate someone will come along and correct me!