Am I doing this wrong? My goal here is to set up a constructor that I can test. The first bit is how I would normally create this new constructor MyObject... the second bit shows how I have to change it in order to test some of the internal functions. See my first comment below.

js-testing.js

//this constructor, MyObject is supposed to only have 1 publically exposed method: .getResults()

;(function ( $, window, document, undefined ) {

  window.MyObject = function(params){
    var error   = params.error;
    
    function _shouldFireRequest = function(){
      //some logic
      return boolean
    }
    
    function _requestResults = function(){
     //makes a request
    }
    
    this.getResults = function(){
      if(_shouldFireRequest()){
        _requestResults();
      } else {
        error()
      }
    }
    
  }

})(jQuery, window, document);

// if I want to test the _shouldFireRequest method, I need to expose it
// as a method on the object as shown below.

// My strategy is that once I expose it, I can create a new instance of MyObject:
// myObject = new MyObject
// set up some state on that instance, then call myObject._shouldFireRequest()
// and test an expectation on the return value.

;(function ( $, window, document, undefined ) {
  //
  //same
  //
  this._shouldFireRequest = function(){
    //some logic
    return boolean
  }

  this.getResults = function(){
    if(this._shouldFireRequest()){
      _requestResults();
    } else {
      error()
    }
  }
  ///
})(jQuery, window, document);

Thanks so much. This is really helpful. In my case. I DO want to have multiple instances of this MyObject... that means, I will use the prototype strategy.

The main takeaway here, that perfectly answers my question is that instead of this._method, like I was doing, I should put those methods on the prototype.

That way, they're kinda from the main constructor, but more importantly they're exposed be able to test in isolation.