Opera stores it's passwords in an encrypted fashion to the phones internal storage, so they should be okay. Granted if your phone is rooted, any app you give root access to can do anything it wants on your phone. Similarly, there are key loggers for phones somebody could potentially install on your phone if they have physical access to it, but this is not really much of a concern.
For other apps, anything saved on the SD card can be read by any other app that has access to the SD card. So you shouldn't store any particularly sensitive stuff there.