How to display taxonomy term custom meta (using wp_get_object_terms?)?

If those extra fields are saved on Edit User page you should get them with get_user_meta function:

$social_twitter_handle = get_user_meta( $curauth->ID, 'firm_social_twitter_handle', true );

If those extra fields are attached to each term you can use get_term_meta function:

$firm_terms = wp_get_object_terms( $curauth->ID,  'firm' );

if ( ! empty( $firm_terms ) ) {
    if ( ! is_wp_error( $firm_terms ) ) {
            foreach( $firm_terms as $term ) {
                echo '<a href="' . get_term_link( $term->slug, 'firm' ) . '">' . esc_html( $term->name ) . '</a>';
                $firm_social_twitter_handle = get_term_meta( $term->term_id, 'firm_social_twitter_handle', true );
                echo $firm_social_twitter_handle;
            }
    }
}