Does the person who introduces me need to use itogai?

Paths shows two groups: the people you know (1st: I know) and the people who can introduce you (2nd: can introduce me). A bridger appears in the second group when they use itogai and have turned on network sharing, and only then can you ask them for an introduction through itogai. Your first-degree connections show either way, so you can always reach out to people you know directly.