Weston, 24, has signed a two-year deal, while 32-year-old Barrett has signed for one year with an option for a 12-month extension from Bournemouth.
Both are available for the Gills' first game of the season against Bradford at Priestfield in League Two.
Gills boss Allen told the club's official website: "Adam has had a tremendous career and has experienced three promotions as a player and captain.
"I have watched Myles many times and he adds another dimension with speed and ability to play in wide areas.
"We've been after both of these players for several weeks now. They both have good experience of getting promotion from this division and with these two we get leadership, goals, experience, speed and excitement."