Rock-Paper-Scissors with the revealing module pattern
I like your question,


I think if you are going to export one function, and then immediately execute it, then you are better served with a self executing query.
You are writing a game, I would use a revealing pattern for the model, the controller, and the view.


View would export 


An alert function
An input function

Controller would export a


init or start function
It would have a lot of internal functions calling view and model

Model would export a


Getter/setter for victories
comparePlays function


Always consider if you can replace control structures with data structures:

// Display a message depending on round result
function privateAlertRoundResults(results) {
  var resultsMessages = [ 'Draw !', 'You lose !','You win !'];
  alert( resultsMessages[ results ];
}


instead of

// Display a message depending on round result
function privateAlertRoundResults(results) {
  switch (results) {
    case 0:
      alert('Draw !');
      break;
    case 1:
      alert('You lose !');
      break;
    case 2:
      alert('You win !');
      break;
  }
}


For extra points, consider how you can convert this into a data structure:

    if (robotPlay == 1 &amp;&amp; humanPlay == 3 || robotPlay == 2 &amp;&amp; humanPlay == 1 || robotPlay == 3 &amp;&amp; humanPlay == 2) {

Consider wellnamed variables or constants instead of magic numbers, your code would be so much more readable if you had upfront

var DRAW = 0,
    LOSE = 1,
    WIN  = 2;


and then later

function privateUpdateDB(check) {
  switch (check) {
    case LOSE:
      privateGameCount++;
      break;
    case WIN:
      privateGameDB++;
      privateGameCount++;
      break;
  }
}


of course in this particular case, I would not have gone for a case but an if:

function privateUpdateDB(check) {
  if( check == LOSE ){
      privateGameCount++;
  } else if ( check == WIN ){ 
      privateGameDB++;
      privateGameCount++;
  }
  //We don't increase privateGameCount for Draws (?)
}

I like your adjusting prompt in case the user provides a bad number
On the whole, prefixing your private functions with private is not a great idea because it interrupts the reading flow. I would not prefix at all, but if you insist on prefixing I would prefix with _, most developers will know what you mean (and frown because they also don't like prefixing ;)