White, 26, has 22 international caps and will join the Premiership club from French Top 14 side Montpellier.
He has also previously played Super Rugby in his homeland for Brumbies.
"When a player of Nic's quality becomes available, and he shows an interest in coming to the club, you have to act quickly," said Exeter boss Rob Baxter.
Back row Freeman, also 26, started the Championship campaign as a vice-captain at Jersey, but has been promoted to skipper since an injury to Pierce Phillips.
"I personally think he's one of the stand-out contributors in the back row in the Championship, and he potentially fills a slot in a Premiership side," Baxter added.
Exeter have signed Australia scrum-half Nic White and Jersey Reds' captain James Freeman on two-year contracts, starting from next season.