1. If you have guaranteed internet access then you can create a shared folder on either Dropbox or Google drive. Either of these has the great advantage of being able to access the files on any device and the great disadvantage of being useless without an internet connection. Neither Dropbox nor Google drive keep offline copies of files on Android. PS in my opinion Google Drive is better for folder sharing than Drop box. It is slightly more fiddly to set up but if gives you more control - you can set up read only sharing for example.
2. If you don't have internet access then things get a lot trickier. You could share a folder in Dropbox and use Folder Downloader for Dropbox to create separate offline copies. this does work but if both users are on the same tablet you are using twice the storage space and the two folders will not remain in sync if anyone makes changes.
3. The holy grail of file sharing is a common folder that both users can access. Android doesn't advertise such a feature but am XDA contributor called BigJArm found one called android/obb. This is a system folder that every user has read write access to. If you stick a folder in there everyone using the device can use it. Here are some quick instructions on how to set this up:
1. WARNING: This is an unsupported, undocumented use of Android so use at your own risk. I have not encountered any issues myself but I recommend making sure that any files you put in here are duplicated somewhere else just in case that directory gets deleted or otherwise rendered inaccessible in any updates to Android. According to this post an OBB is an obscure binary blob, an encrypted package of files that some apps use for storing data.
2. You will need a file manager to do any work on files so grab ES File Explorer from the Google Play Store.
3. Navigate to sdcard/Android/obb.
4. Ignore all the stuff in there and create a new folder using the "+" button at the bottom of the screen. Why not call it "Shared Folder"?
5. Copy or move any files and folders you want into this new shared folder and every user on the device will have read write access to them.
6. I recommend that each user creates a desktop shortcut to this shared folder using ES File Explorer as explained in this tip: How to put a folder short-cut on your home screen
Note: I don't have a good method of sharing apps between users on the one device even though I know this could be useful. For free apps you may as well just download multiple copies from the store. I do not recommend adding your account to someone else's log in just to share paid apps. This will give that person full access to everything on your account including email, calendar and Google drive. If you are going to do that why bother using a separate log in? One approach that might work to share paid apps like games with family members would be to create a brand new Google account for the family and use it to purchase apps you want to share. You could then add this account login to each family member's login via the Android accounts menu under Settings (edit: but don't because you would be breaking the terms of Google Play - see below) . Android is nice that way, it allows you to be logged into multiple Google accounts at the same time. Of course this could leave you open to your kids racking up bills on your credit card and it may be contrary to Google's terms of service. I haven't tried it myself so I do not know.
Sharing. You may not use Products as part of any service for sharing, lending or multi-person use, or for the purpose of any other institution (including, without limitation, libraries), except as specifically permitted and only in the exact manner specified and enabled by Google (for example, through "Social Recommendations").
Note this post is part of a short series of Android tips on working with files and folders. I will be travelling for a while with patchy internet and I needed to get more intimate than usual with the Android file system to make sure I can still access the stuff I need while I am away. I have decided to post the tricks I find for my own future reference and for others who may need to do the same. Separate posts used for each tip to facilitate searching: Android Tips