Send data from Lightning Web Component (LWC) to a Visualforce page

You implemented correctly in LWC of dispatching event. However callback function is not meant for adding some event handlers. You should add event listener directly. Below is the code with changes needed in vf page:

<script>
document.addEventListener("dosearch", function(event){
    console.log('vf event data window listener => ', event.detail);
});

$Lightning.use("c:NotificationSearch_ltngoutApp", function() {
    $Lightning.createComponent("c:lwcNominationSearch",
                               {},
                               "rcontainer",
                               function(cmp) {}
                              );
});
</script>