It's also up to the app developer to some degree. I've had it go both ways, depending on the magazine. Some just download, other's give you a notification and you have to open the app to download. The magazine's cover should change in the Newsstand app/folder, whether the new issue is downloaded or not.
In general, magazines with relatively small file sizes tend to download. The ones that are pretty much just digitized versions of paper publications are often huge. You don't want them automatically downloading unless you have space to spare on the iPad.
If you are getting automatic downloads and don't want them, you can go to Settings > General > Background App Refresh, find that app, and turn it off. That should work, though I won't guarantee it.