With that said, I do use that public sensitivity to help me recognize blind spots I may be missing at home. When the children do not handle themselves the way I would like with others, I file that away and purpose to work on that with them the next week at home. If apologies are necessary, we give those right away, but I do not carry on to my child in front of others.
Again, with that being said, I am not afraid to correct my child in front of others if they handle themselves in a way I know, they know is incorrect. I will quickly remind them of what they should do and expect them to redo it correctly. When they were younger I would just say it out loud and in front of whomever, now that they are older, I wait for a quiet moment and try to whisper to them.
Sorry if it sounds like I am talking out of both sides of my mouth, yet, parenting is like that. Wisdom and discernment are paramount. Knowing what a child is capable of and holding them to it one moment and recognizing they are at a loss and need more training the next.