If you subscribe to a periodical on Kindle, each issue of the periodical can only be viewed on one Kindle device, ever.
In some ways, Amazon makes this pretty obvious up front, but in other aspects, not at all.
First, unlike Kindle book content, periodical content has a designated device, and only downloads to that device, and once one Kindle device has downloaded one issue, other Kindle devices can’t. This is pretty clear in the purchasing UI, e.g. here for The Atlantic, and click “how subscriptions work” near the Subscribe button.
Second, also unlike Kindle book content, periodical content is only available on real Kindle devices, not the software app for Windows, Mac OS, iPhone, Android, etc. (This is supposed to change soon.)
Third, and this is the part that I find non-obvious, once a given Kindle downloads an issue from a subscription, it can’t be transfered to another Kindle. This means that if you break, or lose, or sell, your Kindle, and buy another one, all your old subscription issues are gone. I find this annoying because I didn’t know this when I subscribed — I thought I was buying something to keep as long as I wanted, and of course an individual Kindle device isn’t going to last forever.
Plus, I can’t find anywhere that this limitation is documented publicly (unless you count the terse “Issues are only available for download to one Kindle” statement as covering this case, which I don’t). I found out when I broke my Kindle and bought another one — Amazon makes it easy to transfer the subscription itself to the new Kindle device, so future issues show up on the new device, but it said nothing about the old issues. I then asked Kindle customer support how I could transfer the old issues I paid for, and they said I can’t. Since then, I’ve asked them 3 times if they can point to a public web page with a policy statement on this, and every time they just answer that old subscription content can’t be transfered.
I suspect this comes down to license revocation, and that they never built a revocation capability into their DRM technology, and this is just another one of the lame facts of life about DRM. It’s not the first time that Amazon’s Kindle DRM in particular has reminded people that you’re not really buying content, you’re just licensing it, and the licensing party has undue control over what they’ve licensed you.