Obi-Wan Kenobi both speaks and understands Shyriiwook.
From the script for A New Hope:
Ben is standing next to Chewbacca, an eight-foot-tall savage-looking creature [...] He is a two-hundred-year-old Wookiee and a sight to behold.
Ben speaks to the Wookiee, pointing to Luke several times during his conversation and the huge creature suddenly lets out a horrifying laugh. Luke is more than a little bit disconcerted and pretends not to hear the conversation between Ben and the giant Wookiee.
From the novel:
All but old Ben - Ben, who was talking to the Wookiee in its own language, quarreling and hooting softly like a native.
In The Force Awakens, Rey also understands it.
As shown in this answer from @phantom42 on another site, Rey has at least some understanding of Shyriiwook. Quoting from the novelisation of The Force Awakens:
She tried to find something to say to that, something worthy of the sentiment and the risk they had undertaken. She failed miserably. Chewie, however, had something of his own to add. Whatever the Wookiee had uttered caused tears to well up in her eyes. Having never found himself in such a position before, Finn was unsure how to respond. Knowing well her inner toughness, he wondered what Chewbacca had said that could have inspired such a reaction.
“What’d he say?”
She sniffed and wiped at her face. “That it was your idea.”
Less canonically, Sabine Wren
also speaks Shyriiwook
(so presumably understands it too).
"...it's rough on my throat, I lost my voice for a day after bugging Zeb by serenading him in Shyriiwook."
-- Sabine Wren, Star Wars Rebels: Sabine, My Rebel Sketchbook