My country, Indonesia, is similar to Laos in that it doesn't allow dual citizenship. I don't have to inform the Indonesian embassy that I am now an American citizen and I can still use my Indonesian passport to travel. The complication arises when it comes to entering and leaving Indonesia. Indonesian immigration has, the majority of the time, asked me for my green card or USA visa when exiting. Now that I am a citizen I obviously have no green card to show, and that would give away that I am American. Laos may be similar in that if there is exit immigration checks, they will find out.
Some people I have known use a third country to try and avoid this. Such as leaving and entering Indonesia with an Indonesian passport but arriving/going to a visa free country, such as Singapore, before traveling onwards to the US with their US passport.