The full version of Paperless has a feature to automatically backup your lists, and sync them between devices. (If you're using the free version, Paperless Lite, you can transfer your lists to Paperless instead.)
Turning on automatic backup and syncing
To turn on automatic backup and syncing, go to the app settings screen and tap the "Automatic Backup And Syncing" button:
Then tap the "Link With Dropbox Account" button:
If you already have the Dropbox app installed on your device, it will launch. Simply press the "Allow" button to allow Paperless to connect to your Dropbox account - after which Paperless will reopen and start backing up and syncing your lists. (or, if you'd like to use a different Dropbox account, tap "Use a different account")
If you don't have the Dropbox app installed, a web browser page will open and let you either enter your Dropbox account login information, or set up a new account. Once you've logged in to your account, you'll then be asked to Allow Paperless to access to Dropbox.
Notes on syncing
After you've linked Paperless to a Dropbox account, all of your lists will be uploaded to a "Paperless" folder on Dropbox.
If you link another device to the same account, your lists will automatically be synced between the two devices.
Syncing in Paperless was designed to happen automatically. There is no sync button to press, the app just updates itself as you use it.
Each time you modify a list, or create a new list, those changes are uploaded to Dropbox.
Paperless checks to see if there are new versions of lists:
To ensure proper syncing, lists cannot have the same name. If you have a "Groceries" list on one device which is synced with Dropbox - and link a second device that also has a list named "Groceries" - the two lists will automatically be merged into one.
Unlinking from Dropbox
If for some reason you decide to turn off syncing - or if you wish to connect to another Dropbox account, you can go back to the "Automatic Backup And Syncing" settings, and tap the "Unlink From Dropbox Account" button:
From there you can decide whether you want to: