I've had my ileostomy for 11 years (next month). I've flown a lot in that time. I always pack my ostomy supplies in my carry on because I can't afford to have them lose my luggage if I check it!
As for the way security acts, they're all different, obviously, and I don't want to scare as much as prepare you for worst case scenarios and a tip or two. Honestly, I've had some horrible experiences with security. According to their own website (at least when this happened several years ago) they're not allowed to ask you to show them your ostomy bag. The first time I flew, they saw the bag in the full body scanner, made me walk barefoot (they wouldn't give me my shoes or any of my possessions), to a small room where 3 female security guards interrogated me and demanded I show them my bag. They said if I didn't, they would not let me fly. Despite the fact that I had a print out of their own policies. I cried and cried, it was so embarrassing. They didn't just want me to show my ostomy, I got a full pat down. I had just had the surgery and was still mentally adjusting.
The other times were less traumatic. If I was selected to go through a medical detector, I was passed right through (so if you're given the option, definitely take that). If I was chosen for the full body scanner, they can see it's there so they always pat me down, sometimes run a wand over me or even swab my clothes for explosives residue. I try to stay upbeat and joke with the ladies who pat me down. I learned a long time ago: if you don't laugh, you'll cry.
As for my items, I precut everything in case they take my scissors, which luckily, they haven't. But I have a special prescription steroid foam in an aerosol can (my stoma is a problem, leaks constantly, ruins my skin, bag must be changed everyday). Luckily, when I packed my supplies, I grabbed a new bottle, still in it's unopened box, with the prescription label on it. Because they were giving me a hard time about letting the steroid foam onto the plane, they swabbed it and claimed it had explosives residue on the box! Of course I was shocked and PISSED because it did NOT have explosives residue! After a supervisor checked it again, they said that I could take it because it was in a sealed box with the prescription label on it.
One other tip, you're allowed to bring a bag of medical supplies (ostomy counts) and it doesn't count as your carry on! You get to bring it as a freebie since it's a medical necessity.
I hope I didn't scare you, just hope you can learn from my experiences and have a happy trip!