dynamically call local function in javascript

You cannot get a reference to a local variable by a string. You have to add the local objects to a namespace:

(function(window,$) {
  // Use "var MyObject = " instead of "MyObject = "!! Otherwise, you're assigning
  //  the object to the closest parent declaration of MyVar, instead of locally!
  var MyObject = (function($) {
    var obj = {};
    var local = {};  // <-- Local namespace
    obj.publicMethod = function(number,otherarg) {
      local['privateMethod'+number].call(this, otherarg);
    };

    var privateMethod1 = local.privateMethod1 = function(arg) {
      //do something with arg
    };

    var privateMethod2 = local.privateMethod2 = function(arg) {
      //do something else with arg
    };

    return obj;
  })($);

  window.MyObject = MyObject;
})(window,jQuery);

The private functions are local variables and not part of any object. So, the [...] notation for accessing a property is never going to work since there is no object the private functions are properties of.

Instead, you could make two objects: private and public:

var public  = {},
    private = {};

public.publicMethod = function(number, otherarg) {
  // `.apply` with a fixed array can be replaced with `.call`
  private['privateMethod' + number].call(this, otherarg);
};

private.privateMethod1 = function(arg) {
  //do something with arg
};

private.privateMethod2 = function(arg) {
  //do something else with arg
};

return public; // expose public, but not private

I'm surprised that incorrect answer is marked as accepted. Actually you CAN get a reference to a local variable by a string. Just by using eval:

(function(window,$) {

    MyObject = (function($) {
        var obj = {};
        obj.publicMethod = function(number,otherarg) {

            // Gets reference to a local variable
            var method = eval('privateMethod'+number);

            // Do with it whatever you want
            method.apply(this,[otherarg]);
        };

        var privateMethod1 = function(arg) {
            //do something with arg
        };

        var privateMethod2 = function(arg) {
            //do something else with arg
        };

        return obj;
    })($);

    window.MyObject = MyObject;
})(window,jQuery);

Actually this code is very bad and in 99.9% cases you should not use eval. But you must know how it works and what you can do with it. I myself had a few very specific cases when usage of eval was necessary.