For pinyin representation of Mandarin, these are very different sounds, while the equivalent (identical) Mandarin pinyin representation of し, じ, つ would be xi, ji, cu. I'm not as familiar with romanization systems closer to Latin pronunciations, but for Wade Giles it would probably be written like shi, chi, tsu.

Not exactly. In the Wade–Giles system:

xi → hsi ji → chi ci → tz'u