When I chat using romaji, Japanese will always use si instead of shi, for example yorosiku instead of yoroshiku, siteru instead of shiteru. Do Japanese actually pronounce shi as si?
It's the difference between Hepburn romanization and Wapuro romanizaton. Adults generally use Hepburn system for person names, signboards, posters and such (し =
shi, づ =
zu, じゃ =
ja). But when they need to type ordinary Japanese sentences using romaji, they (almost unconsciously) use wapuro style because it's more efficient and unambiguous (し =
si, づ =
du, じゃ =
ja). You don't have to type
shi to get し when you can get the same character by typing
Please remember that wapuro style has nothing to do with pronunciation. That's just an input method, how they type hiragana/katakana on keyboards. Don't assume anything about pronunciation based on English alphabet.
For their own perception, yes, from the perspective of English speakers, no.
In languages there is a difference between what is called "underlying phonology", which is the discrete sequence of sounds that speakers perceive, and "surface realization", which is what they're actually factually saying. In every language, a process occurs called co-articulation, where sounds take on elements of the sounds that follow them, such as position of the tongue and lips, effectively meaning that sounds blend more into each other in practice.
The "i" is a palatal vowel, a vowel where the tongue is positioned near the palate of the mouth, so while Japanese speakers are pronouncing what they consider "si" the tongue is already moving towards the palate when pronouncing the "s", this results into the s having an objective quality that places it somewhat in-between the sounds of English "s" and "sh" — the end result being that English speakers hear "shi", whereas Japanese speakers just hear "si".
In reverse, the vowel in "gal" is also a palatal vowel, but the vowel in "girl" is not, the result is that the "g" in both words is slightly different with the "g" in "gal" palatalized. This is almost completely imperceptible to English speakers, but Japanese speakers pick up on this, so "gal" got loaned as "ギャル" ["gyaru"] whereas "girl" became "ガール" ["gāru"] — note the "y" in the former.
This gives rise to two different widely used families of Romanization: the "Japanese styles", which are used in Japan, and sometimes in technical settings outside of it, which are based on the Japanese's own perception of their language, and the "Hepburn styles", which are based on the perception of North American English speakers, and mostly used outside of it.
In practice, in Japan, both styles are used, sometimes often in the same word, because Japanese speakers don't know what is up any more; the Japanese government recommends the Cabinet-style variant of the Japanese-style be used for all inter-Japanese communication, which they are taught at school, but also advises Hepburn for external communication with Anglo-Saxons; one can sometimes see "si" and "shi" in the very same word in the wild, due to sloppiness or really inconsistent Romanization.
It's however important to remember that every language has its own phonology and sounds; Japanese is not pronounced like English, or in reverse, and has different sounds. You may look at the Wikipedia article on Japanese phonology; the transcriptions inside of /slashes/ repræsent the underlying phonology that Japanese speakers perceive and what is inside of [square brackets] repræsents the factual sounds made. So what is written as "Fujisan" in Hepburns-style, and "Huzisan" in Japanese-style is perceived as /huzisaN/, with /N/ being different from /n/, by Japanese speakers, which explains why it is written as "Huzisan" in Japanese-style romanization.