To have the right to change the owner for content you obviously have to have access, but you also have to meet one of the following conditions:
be the author of the content in question
have the right of "contributor" to the space or more
have the right of system "coordinator" or more
It is possible to change permissions and visibility of a shared Story.
In the dialog window you can find under the tree dots in your story, you can
See in which space the content is currently published
Change the owner
Grant or remove access to read this Story for one or more specified user(s)
Identify in which spaces your story has been reshared and delete those reshares if needed
Why change owners?
The owner is accountable for the content and is the main contact person towards others
His/her name will appear at the top of the story
He/She is responsible to keep the content accurate and up to date