Denver announced the transactions on Tuesday.
LaCosse was signed off of the New York Giants' practice quad. The 6-foot-6, 255-pound pass catcher had three catches for 22 yards in five games for the Giants from 2015 through 2017. He originally joined the Giants as an undrafted college free agent in May 2015 out of Illinois.
LaCosse appeared in three games and had one start this season for the Giants. He served on the Giants' practice squad since Nov. 14. He spent the entire 2016 season on injured reserve after appearing in two games during his rookie campaign. LaCosse will wear No. 83 for the Broncos.
Leary joined the Broncos this offseason as an unrestricted free agent from the Dallas Cowboys. The 6-foot-3, 324-pound offensive lineman started 11 games at right guard in his first season with the Broncos. He was sidelined for the last two weeks due to a back injury.
Connor McGovern was filling in for Leary. He is the only backup right guard listed on the Broncos' unofficial depth chart. Denver has Max Garcia and Allen Barbre listed at left guard and Donald Stephenson, Barbre, Elijah Wilkinson and Cyrus Kouandjio listed at right tackle. Garrett Bolles and Stephenson are listed at left tackle.
Denver has Virgil Green, Jeff Heuerman and Austin Traylor listed at tight end.
The Broncos battle the Indianapolis Colts Thursday at Lucas Oil Stadium in Indianapolis.