It is fairly easy to copy the source information (and language code information) from one copy of translationStudio Android to another. However, I recommend backing up the translationStudio folder beforehand, and this is probably best used on a fresh install.
First, make sure you have the necessary source text files installed for your intended gateway language. In my case, it was Swahili. I needed to first update the list of Gateway Languages, and then I needed to download the actual files. The Old Testament, New Testament, and Open Bible Stories are all done separately, and each book of the Bible is a separate download (although you can select all by tapping the check box above the list of books, and download one whole testament together).
Once you have confirmed that you can create a project with source text in your chosen language (or create a project with a particular language code) you will need to get some files over to the other tablet(s). If you are working with multiples (I have six to do) I advise putting this on a thumb drive. If you are going to update only one you can feel free to use ShareIt or SuperBeam. There are a lot of files involved, and it will take time.
Assuming that you will use a thumb drive, go into your File Manager. The translationStudio folder should be at the root level (base level) of the file system on your tablet. Inside the translationStudio folder are two folders you want: databases and resource_containers. Long-press these and select and copy them. You may have to select Show SD Card depending upon which File Manager you are using. Go to the USB drive you have connected, and paste the folders. I put them in another folder to keep things tidy.
Now, eject the USB drive properly and connect it to the other tablet. If translationStudio has never been used on this tablet, it will be helpful to run it once. You can quit when you get to the sign-in screen. (You don't need to sign in, even as a local user.) After this, there will be a translationStudio folder in the file system of this tablet.
Copy and paste the two folders from the flash drive to the translationStudio on the tablet. You should be asked if you want to overwrite the existing data, and you should confirm this. When you are done, inside the resource_containers folder there should be a folder starting with the Gateway Language language code for each book of the Bible supported (in my case, sw).
You should now be able to create translation projects using the copied source files.
