option1.js

function MyThing() {
    var theThing = new Thing();
    theThing.privateMethod();
    return theThing;
}

// Somewhere else...

var whatIWant = MyThing();

options2.js

function MyThing() {
    Thing.call(this);
    this.privateMethod();
}

// Somewhere else...

var whatIWant = new MyThing();