Set the correct privileges in the catalog: being allowed to open and edit an existing object doesn't implicitly means you will be allowed to overwrite it with your changed. Often that kind of security is used when providing some "standard" reports and users are allowed to take them, change them and store them in a different path (often their own personal folder).
For objects in the catalog you have a "write" permission, that's the one allowing you to overwrite an object (or store a new one in a folder).
Thanks for your response, but the issue continues to persist.
So, users were initially able to create and save all of their work (dashboards and analyses) in the past, so they have always had the necessary privileges. This issue randomly started occurring a few days ago and it's impacting multiple users now.