As of now there is no direct option to save PDF locally from Adobe Acrobat app on iOS.
But this is what you can do.
Open Safari browser
Go to https://cloud.acrobat.com/ & sign in with your Adobe ID
Select the PDF that you would like to download
This will open the PDF in Safari
Tap in the PDF and this will give you an option 'Open in...' at the top right.
Select "Copy to Adobe Acrobat".
Open the PDF document in Adobe Acrobat app
Tap the Share icon at the bottom right
Select Share File from the Popup
Select "Share Original Document" and tap 'Mail'
Email it to yourself
Open the email attachment in Adobe Acrobat app.
Once the PDF is opened in Adobe Acrobat app for iOS, it is stored locally.
Thank you. The need to do this is ridiculous, IMHO, and is a conspicuous attempt to make the Adobe Cloud "sticky". One did not always have to go through this within the Adobe mobile app, and the gyrations now required to avoid Data charges when accessing your documents are a waste of time..
I found a better solution to be to download all the files in my Adobe Document Cloud to my PC and save them in another Cloud service that does allow you to download them directly to your mobile device. I can then save them locally in Acrobat Reader or another app as needed.
Why do you do this to us Adobe? This is such a lame and frustrating process! Come on.