The highly touted quarterback, picked No. 22 out of Oklahoma State, joined No. 3 overall pick Trent Richardson, the former star running back at Alabama.
Pat Shurmur opened his second season as coach in Cleveland on Wednesday.
Weeden, who signed a four-year, $8.1 million contract Tuesday, is expected to get the bulk of work. The 28-year-old rookie, a former minor-league pitcher, will be trying to win the starting job from McCoy.
Weeden and McCoy came out together, throwing the ball back and forth before actual drills began.