Hyde received a five-year, $30.5 million deal with $14 million guaranteed.
The 26-year-old Hyde has played both safety positions and cornerback during his four-year career. He recorded three interceptions last season and has eight in 63 career games (33 starts).
Poyer missed the final 10 games of last season when he suffered a lacerated kidney. The 25-year-old free safety started all six games prior to his injury and has two career interceptions in 48 career games (10 starts).
Buffalo also signed kicker Stephen Hauschka, guard Vladimir Ducasse and fullback Patrick DiMarco. The team also released safety Aaron Williams.
Hausckha, 31, spent the past six seasons with the Seattle Seahawks. He was 33-of-37 on field-goal attempts last season and has converted 87.2 percent over nine NFL seasons.
DiMarco spent the past four seasons with the Atlanta Falcons. The 27-year-old has 37 career receptions for 273 yards and four touchdowns. The Bills coveted DiMarco even after signing free agent Mike Tolbert earlier in the week.
Williams suffered a season-ending neck injury in each of the past two seasons. The 26-year-old doesn't yet know if he will get to resume his career. He had seven interceptions in 59 games (52 starts) over six seasons with the Bills.