How excately does the revealing take place.
As ringkichard stated, this is not really something covered by the rules. It's more an etiquette question than a rules question if you ask me.
I personally play it similar to ringkichard. If I want to reveal an enchantment I'll declare my intention to my opponent, such as "Wait, before you continue I would like to reveal this enchantment." It is then up to my opponent to state an intention to also reveal an enchantment. If they fail to do so before I have revealed mine, then it is too late for them to do so.
Now if you use this method, courtesy dictates that you do give your opponent time to respond. You don't just go "Aha!" as you flip over the enchantment. But again, it is entirely an etiquette thing that you have to work out individually with your group. Some people may be more comfortable asking their opponent if they would like to respond rather than risking hurt feelings from people who don't think you gave them ample opportunity to respond.
Whatever method of common decency you decide to personally abide by, however, I do not believe you would ever be required to reveal if your opponent beat you to it. The rules on pages 18 and 19 of the Rulebook v2.0 clearly make use of the word "may." Basically what happens, is you both declare a
desire to reveal an enchantment. The person with initiative gets the first opportunity to actually act upon that desire and reveal their enchantment. They are not required to. Then the other person gets the opportunity to act upon their desire and reveal. If the actions of your opponent has made your desired action no longer beneficial, you can certainly change your mind and decide not to perform it.