I forget which fic I pulled this headcanon from, or if it just kind of bubbled up as collective fanon thing, but I always imagined that Naruto was sending stuff back to Konoha either via toads or regular slow civilian mail. But, like, months off Small things–like a postcard or something from one country when they’re already two countries over, or sent along with a merchant caravan that will end up in Konoha eventually.
Naruto didn’t grow up with a lot of bonds, remember, so he wouldn’t really know the culture of gift giving and letter writing so much. A lot of his friendships are in the moment, in person–not that he no longer considers them friends when they’re long distance, but it’s not something I imagine him being so conscious of as to send gifts?
I mean, then again, this is all minutiae that Kishi never bothered with, but who is in Konoha that he’d want to send stuff back to in canon? Sakura and maybe Iruka and Tsunade? Possibly Kakashi, though what besides snippets of Jiraiya’s books would he want?
Contrast, of course, with DoS where Shikako did actively cultivate stronger bonds amongst her friends. Deliberately made post-mission team dinners a tradition, and fostered a sense of shared stuff amongst her team.
And I do think Tsunade and Jiraiya keep each other appraised of the respective situations to some extent–now, whether Jiraiya feels like sharing that is up to him.