You can use "Open in" option to open file on any other application on your device like iBooks. In this way the PDF will be associate with that application and will be saved on device.
Refer this KB doc for help : Acrobat Reader Help for iOS: Emailing and printing
Let us know if you need any help.
What about copying an entire folder for local? We meet with clients and have 8-10 forms in their folder to review. For years we used PDF Expert for it's better file handling and offline abilitie. Adobe-Dropbox does not work offline and has no mechanism for copying a Dropbox folder to "local" and then back to Dropbox. Not sure what subscribing would add in features but I do know my staff is not paying $25 a month for what they have for free in PDF Expert
Have we missed something? We do prefer the integration of Dropbox/Adobe.