|Yes, you are correct. If the data is sensitive then snap the disk into small pieces or score the 'label' side to make it unreadable.|
CD-R (& CD+R) disks are designed to be written to once and, as such, are a good way of keeping a (relatively) permanent copy of data but they are no good for use as a 'use many times' medium, for that you need the ones with 'rw' at the end (for 're-writable'). The same goes for DVD and Bluray disks.
"I've always been mad, I know I've been mad, like the most of us..." Pink Floyd