Specifically, small roughly 3" square-shaped drawers that are partially enclosed on top, while the sides curve down towards the back, all the way down to nothing. You can see one in this video at around the 2:12 mark.
I have heard them called “tea drawers” (or “T drawers?”) but I’m sure they aren’t for tea, which would all spill out, and why would you store tea in a desk?
I have my own WAG, which is this: they were designed to sort and store private correspondence or other documents, which would require that they be pulled all the way out and turned around for sorting, and when they were put back in the desk the correspondence or documents would be hidden from view. They were not secret, just identified by this arrangement as “private,” enough in those days to keep honest people from prying.
There were six such drawers in this desk, plus a number of slots and pigeon-holes for sorting papers that didn’t need to be hidden from view. I can imagine someone in business needing that kind of sorting capacity, but it does seem a very roundabout way of getting privacy for some of one’s documents.
My original thought would be that they’re designed like that, with the swooping sides, so you could tilt them down instead of taking them out (or standing up) to see everything. But that wouldn’t explain the partial cover and no back.
My guess is that they were meant to be completely removed, turned 180, and placed on the desk surface while the stuff in them was being used, then lifted and replaced when done (sort of like those little “desk organizers” they have today) to get them out of the way. (i.e. the same guess as the OP, except for convenience rather than privacy).
I have no evidence for this guess other than “that’s what I would do with it.”
As I look at the video a bit more, that’s sort of what I’m thinking as well. It seems like it would hold envelopes or writing utensils in such a way that they won’t fall out while you’re moving as long as you have it tipped a bit and the partial cover makes sure something springy (like a stack of envelopes) doesn’t make the drawer impossible to open.
But I couldn’t find anything like it elsewhere on the internet. Granted, I know no woodworking terminology and I had the video muted so I had to take the OPs word for it being t or tea drawer.
I think Roderick Femm is correct in that they are for letters. The wide drawer underneath each set of three is for full, unfolded sheets of paper. The smaller drawers are for sheets of paper folded in thirds as you would for mailing. I suppose envelopes would fit in there as well. Because the drawer is closely matched in size to the papers inside, they have no back to facilitate removal of the contents.
Sometimes old commercial desks like that would have slots on the back, and the desk would be outward facing so that people could walk by and insert documents, which would then wind up in those drawers.
This desk doesn’t have those slots on the back, but the drawers could be shared with models that do for cost reasons, and because they make it easier to grab stacks or flip through them once the drawer is out, could have been a common convention for any small document drawers.
So I agree with the others above: They’re probably small document drawers.