<h:form>
    <ace:pushButton id="button1" action="#{someBean.someFunction1}" label="Button1" />
</h:form>
<h:form>
    <ace:pushButton id="button2" action="#{someBean.someFunction2}" label="Button2" />
</h:form>

function setTestButtonsOnclickFunctions() {
    var button1 = document.querySelector('[id*="button1_button"]');
    var button2 = document.querySelector('[id*="button2_button"]');

    button1.onclick = function() {
        button2.click();
        var someRandomVariable = 10;
    };

    button2.onclick = function() {
        button1.click();
        alert("Button1 was clicked from javaScript");
        var someRandomVariable = 10;
   };
}

(function($) {
    $(document).ready(setTestButtonsOnclickFunctions);
})(jQuery);