User confirmed recreation solved the problem. It is behaving normally again for him. I'm not sure how to indicate the status change in this new Communities environment.

I am thinking the definition of the analysis object must be corrupt. I recreated the 2 reports affected and the new report appears to be behaving normally (able to edit and save or save to the test directory or My Folders). I'll write back here on Monday with the outcome once the user has a chance to test on his own.