As i see you are trying to rewrite the model.so you have to have rewrite the model in the following way

<models>
    <catalogsearch>
        <rewrite>
            <resource_fulltext>Sid_Newsearch_Model_CatalogSearch_Resource_Fulltext</resource_fulltext>
        </rewrite>
    </catalogsearch>
    <newsearch>
        <class>Sid_Newsearch_Model</class>
        <resourceModel>newsearch_mysql4</resourceModel>
    </newsearch>
</models>

As i see you are trying to rewrite the model.so you have to have rewrite the model in the following way

<models>
    <catalogsearch>
        <rewrite>
            <resource_fulltext>Sid_Newsearch_Model_Resource_Fulltext</resource_fulltext>
        </rewrite>
    </catalogsearch>
    <newsearch>
        <class>Sid_Newsearch_Model</class>
        <resourceModel>newsearch_mysql4</resourceModel>
    </newsearch>
</models>

and importantly you are trying to rewrite wrong module which doesn't exists.your class should be

Your class under Sid/Newsearch/Model/Resource/Fulltext.php should be like

class Sid_Newsearch_Model_Resource_Fulltext extends Mage_CatalogSearch_Model_Resource_Fulltext

try avoiding folder called CatalogSearch under Model.