The 22-year-old Borini, who has one senior cap for his country and was in Italy's Euro 2012 squad, and Dossena become Paolo Di Canio's 12th and 13th signings of a busy summer.
Borini moved to England as a teenager when he joined Chelsea from childhood club Bologna, but he could only manage eight appearances for the Blues and spent time on loan with Swansea.
While at the Liberty Stadium, Borini scored some important goals in their promotion to the Premier League, featuring in their play-off final success.
Borini returned to Italy and spent time with Parma and Roma, where his impressive form was enough to earn a move to Liverpool.
He made 13 Premier League appearances last season but the arrival of Daniel Sturridge and Philippe Coutinho pushed him well down the pecking order at Anfield, and he did not feature in any of their opening three games this season.
Left-back Dossena also has a Liverpool link, having played for the Anfield side from between 2008 and 2010, albeit only making 18 Premier League appearances across his 18 months at the club.
The 31-year-old, who joins Sunderland on a free transfer, left Liverpool for Napoli in January 2010 and was a regular until last season when he only played eight times.