You may see your owned guns either through the Weapon Store or by pressing the Switch Weapon button in Loadout.

The Weapon Store can be accessed from the main Landing Page when you first enter the game and it is sorted by weapon type (Rifle, Shotgun, Assault Rifle, etc.) Weapons you own in each of these category types will appear mixed in with weapons you do not own. In addition to buying new weapons or seeing weapons you currently own, the weapon store will also indicate which weapons you have equipped and allow you to switch to a different weapon for that type. You may have one of each weapon type equipped at any time.

You may also see owned and available weapons by type from the Loadout screen by pressing the Switch Weapon button on the navigation bar at the bottom left corner of the screen. This view will provide you a list of owned weapons that are of the correct weapon type for the mission you are about to play.