To be able to use this feature you have to have the Manage Capptions permission.
Go from your dashboard to Capptions in the menu left and open a Capption.
This example Capption is completed, but you can also transfer a Capption when it is still in progress.
Hit Options in the upper right corner and hit "In progress" for someone else.
Select a user to transfer the Capption to and hit Save.
The Capption will now be available in the environment/app of the selected user.