Correct, this is why I'm wondering. Input is done through hiragana such as "わたし" (or even a step further removed, transliterated from "watashi") and then the IME is triggered to convert it to "私" (or other matches).
On a laptop, my experience is that romaji -> hiragana is as-you-type due to being unambiguous, while the default trigger for hiragana -> kanji is the spacebar, same as described for Korean. Hence my confusion as to how it's different - the individual characters certainly are, but it sounds functionally identical.
On my phone just now, typing this comment, the experience was switching to a Japanese keyboard and inputting the hiragana directly, then the kanji suggestions appeared where autocorrect suggestions normally would.