Full-back Watson, 19, and 20-year-old back-row forward Sisi both featured in this season's Under-20 Six Nations tournament.
Bath have now recruited four players from Irish for next term, with Jonathan Joseph and Matt Garvey also heading to the Recreation Ground.
"Anthony and David are outstanding players with real potential for the long-term," said Bath head coach Gary Gold.
"We feel that by bringing them into this environment we will really see them develop and bring out the best in them.
"Young English talent is a big part of our thinking, and we see these young men as investments for the future. We are delighted to have secured them both."
- Sports & Recreation