Front row Faletau, 28, has three caps for his country and will arrive from Championship club Yorkshire Carnegie.
Fellow prop Cosgrove, 22, is a former Scotland Under-20 international who is also eligible to play for England, and will arrive from the Pro12 outfit.
"The Championship has some excellent players and Sione is in his prime," said interim Bristol boss Mark Tainton.
"Both players were impressed that there's a clear understanding about what we want to achieve, and they're eager to be a part of that."
Bristol are bottom of the Premiership, seven points from safety with five games remaining.
Bristol have signed Tonga international Sione Faletau and Edinburgh's Jack Cosgrove for next season.