You need to tell all of your health care providers and anyone who will be helping you as you recover. No one else really needs to know. The stoma won’t be obvious under your clothing, you won’t smell, and you won’t make any more embarrassing noises than you made before ostomy surgery. When people ask, you can simply respond that you had abdominal surgery to remove part of your colon or bladder. It can be helpful if a trusted co-worker or your supervisor are aware of your ostomy status as a matter of practicality at work.
Many ostomates find it helpful to set up a signal system with a close friend or family member, so that if someone notices a leak they can quickly let you know.
If you have children or teens, they will respond better if things are not made secretive and mysterious. A matter of fact approach and direct answers to any questions are most effective.
Finally, there are some people who simply are not well-informed about ostomies and they may have negative perceptions. A general rule for dealing with this and communicating about your ostomy is: If you believe that your life revolves around your stoma, then that is how other people will see you too. If you put in the effort to cope and adapt, you can just get on with your life.