As you said past issues of Kindle magazines and newspapers can be stored and downloaded from the Cloud. In the case of canceling your subscription for that particular magazine, you can still access them from the cloud. But since the size and delivery frequency of the subscriptions vary making only the most recent issues available for you to view from your device in Cloud.
But if you want, you can save these issues on your Kindle. Generally, when a newer issue comes the older issues gets deleted automatically from the device. So in order to save the older issues, you have to:
Go to Home screen of your Kindle Fire > tap on Newsstand.
Then select the issue which you want and Press and hold them.
Finally, tap on Keep This Issue.