Since you are backing your Select2 with an input element, rather than a select element, and it allows multiple selection, I believe you need to define an initSelection function.

Add the following option:

initSelection: function (element, callback) {

    callback($.map(element.val().split(','), function (id) {

        return { id: id, text: id };

    }));

}

jsfiddle

Note: Instead of calling the following:

$(test).val([test1,test2]).trigger(change);

You could have called this:

$(test).select2('val', [test1,test2], true);

And when you do that without defining an initSelection function, you get the following error:

Error: Error: val() cannot be called if initSelection() is not defined

Note: I think the reason your first (non-ajax) example works without defining an initSelection function is because it specifies the tags option.

UPDATE: Select2 v4

When using Select2 v4, you should always back the Select2 control with a <select> element, not a hidden input.

<select id=test style=width: 300px; multiple=multiple>

</select>

Note: You can still specify multiple: true in the options, but you can also use the multiple attribute of the <select> element.

You can set default values by including selected options in the html or by programmatically adding selected options to the <select> element, after which you should trigger a change-event on the element so its display is updated.

$test.append('<option value=initial1 selected=selected>initial1</option>');

$test.append('<option value=initial2 selected=selected>initial2</option>');

$test.trigger('change');

In this case, the initSelection function is no longer needed (or allowed).

jsfiddle

Note: Select2 does have a full version which supports some backward compatibility. That may also be an option.