With Instagram... you are probably safe as long as you are not sideloading the app from SketchyJohn's App Emporium.
Those app permissions descriptions can be quite misleading... When an app says "Access to SD storage" and "Permssion to acess your picture library" or something... well.. it HAS to.
In order to take a picture, save it to your phone then go in and access that picture at a later date, it needs the access. The permission warning is kind of a 'worst case' scenario thing where they are TRYING to get your attention so you just don't rubber stamp an app's permission.
For instance...some flashlight apps... those are HORRIBLE.... they'll include permissions to access your location, your internal storage, transmit on the network, etc... When you see a flashlight app looking to know your location and get access to the network, THEN worry. But if its a photo program from a well known source, you're in good shape.