How is it possible to remotely access photos on iPhone? Is there any software to do so, and what would it take to install it?
Another person had physical access to my phone, so I suspect the possibility of installation of such software.
Even photos which I immediately deleted and those, which went straight to application got in other hands.

I have an iPhone 4s. It was not jailbroken (at least, by me), and it did not appear such: no Cydia app or other oddities.
Photo Stream and iCloud backup for photos were also off, and I never used them.
Dropbox, Google+ and other similar apps offer the ability to automatically upload new photos to your cloud storage. Do you have anything like that installed?