(To keep things simple, this post has been written in terms of two parent households. The most of the same issues and choices will apply regardless of your family set up.)
The big problem with being closeted is that you can be outed. While some folks managed to stay closeted for decades, that’s the exception rather than the rule. And relationships that do stay closeted long term are usually not trying to hide the truth from people they live with. Sooner or later, many poly parents will need to deal with their kids learning the truth.
Kids who discover you have other relationships will likely be tweens or teenagers. Younger children generally don’t have enough awareness of social conventions and adult interactions to know the difference between their parents having friends and their parents being in intimate relationships. Tweens and teens who discover their parent’s relationships will make their own assumptions about what is going on. At best they may assume or hope that you have an open relationship. At worst, they may assume that you and/or their other parent are having an affair or are on your way towards divorce.
Unfortunately, not all kids will come to you about what they know or assume. This creates a difficult situation–you think your relationships are secret and their assumptions are running wild. If they are assuming affair or divorce, this can put a huge amount of stress on your kids.
Hopefully, if your kids do discover your relationships, they will be comfortable enough to come to you or your partner about it.
You best approach at that point is just to be honest. Make it clear that no one is cheating, no one is sneaking around, and that your kid’s family is not going to be torn apart by your relationships. They may have questions, they may be upset or feel threatened, they may not care once they know that nothing is going to change for them. Give them some time to absorb everything.
If possible, I recommend giving them an idea of who they can talk with. You being in the closet should not deny your kids a chance to have a support system. They need to have a family friend, relative, therapist, someone they can talk with (who isn’t you). Unless you intend to come out, in general, your kids will need to know that you are in the closet and why. That means they will need to know who is and is not safe to talk with.
If your kids do not come to you, you may notice changes in their behavior, mood, and how they treat you or your partner. Of course, that also describes stereotypical teenagers. It is always hard to tell when tweens or teenagers are dealing with a major problem that needs their parent’s intervention, and when they are dealing with the regular stresses of being a teenager.
If you suspect your child has discovered your relationships, but they say nothing to you, you’ll need to decide how to approach them. You might take the head-on approach of outing yourself. This will make it easier to discuss your relationships and any concerns they may have. However, if they hadn’t yet discovered your relationships you may have outed yourself unnecessarily. Or you can try to get them to tell you what is wrong. This is notoriously difficult with teens, and only slightly easier with tweens. If they have discovered your relationships, or if something else is going on that damages their trust in you, it will be even harder.