My Media:
My Media is your personal media storage. It is different for each user because it shows media you yourself have uploaded or created over time. The same “My Media” list is available to you from each course, and items in My Media can be published in one or more courses. Items in My Media are private to you until they are published somewhere. You can also add other users as collaborators on your media from within My Media.
Media Gallery:
The Media Gallery is private to the course you are in. Everyone enrolled in that course can see what’s published to the Media Gallery. The Media Gallery does not necessarily include all media that is used within the course in other contexts. Instead, the Media Gallery contains the content that was deliberately assigned to it. The Media Gallery can be added to the Blackboard Course Menu using the Tool Link
Multiple people in the course can contribute to the Media Gallery depending on the way the admin setup the course.