When double doors don't close properly, it's almost always because the doors don't meet in the middle. The striker on one door fails to penetrate properly into the adjoining door. It's usually caused when one or both doors sag from the top on the hinge side. It can cause the doors to rub together at the top while separating halfway down. The sag is usually the result of a loosening of the hinges or doorjamb, caused by the weight of the door. Another issue might be a worn out striker plate.
Open both doors. Remove the top two hinge screws from the top hinges on both doors using a drill/driver.
Measure the length of the screws. Select replacement screws that are at least 2 inches longer than the original screws.
Place the longer screws in the hinges where you removed the originals. Drive them in with force using the drill/driver. The screws will penetrate though the jamb and into the stud frame around the door to secure the hinge, and drive the jamb back plumb with the wall.
Remove the striker plate from the door that typically remains in place, or the door with the vertical cylinder that locks the door at the top or bottom. The striker plate is the small brass plate on the side of the door with a hole in it.
Screw a replacement striker plate to the door using the original screw holes. Use 3/4-inch screws and the drill/driver. Snug the screws only. Don't tighten them all the way.
Shut the door. If it doesn't work to your satisfaction, examine the striker and striker plate alignment. Tap the striker plate in the direction needed to center the striker on one door in the hole in the striker plate on the adjoining door. Test and adjust as needed until the door closes tight. Tighten the screws to finish.