The sound you hear in HI is not really a "sh" (as the English "sh"), but neither is the sound SHI an "sh". While it's very easy to learn to pronounce Japanese sufficiently, Japanese pronunciation does have its quirks, and you have to get used to it. The "sh" situation (or fricative situation, as we'd call it in linguistics lingo) is one of them.
Let's first consider the sounds in hand. SHI and HI are both just romanized transcription of the Japanese hiragana letters し and ひ. These letters (like most kana letters) represent a combination of a consonant (//h// or //s//) and a vowel (//i//), but in Japanese, some of these consonant+vowel combinations can make the consonant sound change drastically. We usually see it in the common transcription system (Hepburn), so when we combine //h// and //u// we get FU (ふ) and when we combine //s// and //i// we get SHI (し). But in the case of //h//, although the pronunciation changes (and it changes for everyone in the standard dialect — you just don't always get to hear it), the transcription system falls short of representing that change.
Now let's look at the combination //s// + //i//: the vowel //i// causes a sound change that belong a very common class of sound changes called palatalization. These changes usually happen before front vowels (such as //e// or //i//) and this is the origin of the "soft" C and G in English that usually come before an E or an I (this palatalization originally occurred in Latin, so it has equivalent in most European languages since they've all borrowed stuff from Latin, or are direct descendants of it).
In Japanese, the palatalization we speak about occurs before //i// and //j// (not before //e//). //j// here means the consonant that's usually transliterated as Y (since it sounds like "y" in English), in combinations such as RYO (りょ). When this consonant is combined with //s// we usually transliterate it as sh, so しゃ (//s// + //j// + //a//) is transliterated SHA, although you sometimes always find the transliteration SYA.
So how does this changed pronunciation really sounds? To the ears of an English speaker (or most Westerners) this usually sounds exactly like the English "sh" — but it is not. The sound of the English "sh" is pronounced with the tip of the tongue behind the teeth, but pointing toward the palate (the roof of the mouth), while the Japanese SH is pronounced somewhat similarly, but the tip of the tongue is not pointing up, and the "sh"-like sound is actually produced by the having back of the tongue raised against the palate. These sounds are different, and some languages may distinguish between them. As it happens with English in Japanese though, it doesn't matter, and we'd just perceive the first one as "English accent" and the second as "Japanese accent". Linguists mark the first (English) sound as [ʃ] and the second (Japanese) sound as [ɕ].
With that settled, we can move to what happens with HI. In essence, it's very similar to SHI — we also have palatalization that happens in the same way (so it also occurs in HYA, HYO and HYU, although it is not reflected in the transcription), but //h// is palatalized to something slightly different. This consonant is called palatal fricative, and it's theoretically simpler to pronounce — you just raise your back of the tongue against the palate (you don't do anything with the tip, as opposed to the Japanese SH). This sounds a lot like a softer, more fluid version of "h".
Learning to differentiate between all these sounds takes practice, but I suggest listening to the recordings at the Wikipedia articles and trying to see the difference yourself. The articles themselves are highly technical, but they also contain a list of examples from other languages where you may find these sounds.