[Galaxy Tab S Q&A, Help & Troubleshooting] prorokrak:
I'm not sure how the Private Mode works exactly. I just know that when I enter the Private Mode, a new "root" folder is mounted in Samsung's "My Files" file manager. Also, in the "storage" folder a new folder appears, namely "private". When I tried to find the files while the private mode was turned off, I was not able to find them. In my current Lollipop ROM (which has secure storage disabled), there is also a folder named "privatemode" in the "mnt/shell" folder. The "privatemode" folder contains the files that probably appears in private mode. But I can' tell for sure since the names of the files are encrypted and I guess their content is also encrypted.

However, I am not sure if these files were visible also in my KitKat ROM with secure storage enabled. And I don't know if the files could be used (decrypted), if they would be simply copied into the new ROM.

Possibly, but I didn't know where to find them. Now I know I would look into mnt/shell folder, and if there was some content in the privatemode folder, I would give it a try and copied them into the new ROM.

While it is probably true, in the ROM description it is stated that secure storage enabled doesn't work with Xposed Framework. Probably it was also true on KitKat. When I wanted to enter Private Mode, I had to uninstall Xposed, restart the device, do what I needed in Private Mode, install Xposed back and restart to get the framework activated. Not very handy :( Now, with secure storage disabled, I can use Private Mode together with Xposed Framework. :victory:

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Anyway, since I considered restoring backup quite easy and definite way how to get my files back, I have already done it. Since I was at work, I didn't have time to tinker around, but I had a plenty of time to do restore/backup thing. Now, I am already back on my Lollipop, having the missing files restored. Everything works just as before.

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I would like to also share my experience with restoring the backup. Interestingly, after restoring the backup of KitKat system, not everything went smoothly. For example, the internal storage was not properly mounted. I could see content of "sdcard" folder in My Files file manager but other apps (including ES File Explorer with root permissions) saw "sdcard" folder empty. But when looking into the data/media folder, everything was there.

Moreover, when trying to uninstall Xposed Framework in order to get Private Mode activated, after restart I got stucked at Samsung logo, and had to power off the device, boot into TWRP recovery, and manually rename /system/bin/app_process.orig back to /system/bin/app_process (as I learned here http://xdaforums.com/xposed/xposed-installer-versions-changelog-t2714053).

I guess these problems could be caused by not wiping Dalvik Cache and Cache partitions after restore. Otherwise, it would be unpleasant surprise that a system that worked fine prior to backup, would not work so flawlessly after restore. But since I was able to do what I needed, I consider it mainly as an interesting experience.

Once again, thank you very much for your help, expertise and interest to help my solve my problem. It really is a good feeling to see such willing spirit.