Notify on Save only! Plugin demonstration of Customize Setting Validation: validates Customizer settings that contain titles to ensure they are not empty, and then blocks saving the Customizer until they are populated. See https://github.com/xwp/wp-customize-setting-validation

customize-validate-entitled-settings.js

/* global wp, _customizeValidateEntitledSettingsExports */
/* exported customizeValidateEntitledSettings */
var customizeValidateEntitledSettings = ( function( $, api, exports ) {

	var self = {
		l10n: {
			invalid_value: ''
		}
	};

	if ( exports ) {
		$.extend( self, exports );
	}

	/**
	 * Add validation to a control if it is entitled (has a title or is a title).
	 *
	 * @param {wp.customize.Control} setting                   - Control.
	 * @param {wp.customize.Value}   setting.validationMessage - Validation message.
	 * @return {boolean} Whether validation was added.
	 */
	self.addValidationForEntitledSetting = function( setting ) {
		var initialValidate;

		if ( ! self.isEntitledSetting( setting ) ) {
			return false;
		}
		initialValidate = setting.validate;

		/**
		 * Wrap the setting's validate() method to do validation on the value to be sent to the server.
		 *
		 * @param {mixed} newValue   - New value being assigned to the setting.
		 * @returns {*}
		 */
		setting.validate = function( newValue ) {
			var title, setting = this;
			// Note: if we want to get the old value, just do oldValue = this.get()
			if ( _.isObject( newValue ) ) {
				title = newValue.title;
			} else {
				title = newValue;
			}
			newValue = initialValidate.call( this, newValue );

			if ( _.isString( title ) ) {
				setting.validationMessage.set( '' );
			}
			return newValue;
		};

		return true;
	};

	/**
	 * Return whether the setting is entitled (i.e. if it is a title or has a title).
	 *
	 * @param {wp.customize.Setting} setting - Setting.
	 * @returns {boolean}
	 */
	self.isEntitledSetting = function( setting ) {
		return (
			'blogname' === setting.id ||
			/^(nav_menu_item|widget_.+?)\[/.test( setting.id )
		);
	};

	api.bind( 'add', function( setting ) {
		self.addValidationForEntitledSetting( setting );
	} );

	return self;

}( jQuery, wp.customize, _customizeValidateEntitledSettingsExports ) );

customize-validate-entitled-settings.php

<?php
/**
 * Plugin name: Customize Validate Entitled Settings
 * Description: Prevent the site title, nav menu items, and widget instances from being saved without titles. Depends on <a href="https://github.com/xwp/wp-customize-setting-validation">Customize Setting Validation</a> plugin.
 * Author: Weston Ruter, XWP.
 * Plugin URL: https://gist.github.com/westonruter/1016332b18ee7946dec3
 * Version: 0.1
 * Author: XWP
 * Author URI: https://xwp.co/
 * License: GPLv2+
 * License URI: http://www.gnu.org/licenses/gpl-2.0.txt
 * Text Domain: customize-validate-entitled-settings
 *
 * Copyright (c) 2015 XWP (https://xwp.co/)
 *
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License, version 2 or, at
 * your discretion, any later version, as published by the Free
 * Software Foundation.
 *
 * This program is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with this program; if not, write to the Free Software
 * Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
 */

/**
 * Class Customize_Validate_Entitled_Settings
 */
class Customize_Validate_Entitled_Settings {

	/**
	 * Most recent instance of this class.
	 *
	 * @var Customize_Validate_Entitled_Settings
	 */
	public static $instance;

	/**
	 * Translated strings
	 *
	 * @var array
	 */
	public $l10n = array();

	/**
	 * Return a new plugin instance.
	 *
	 * @return Customize_Validate_Entitled_Settings
	 */
	public static function instantiate() {
		self::$instance = new self();
		return self::$instance;
	}

	/**
	 * Customize_Validate_Entitled_Settings constructor.
	 */
	public function __construct() {
		add_action( 'customize_validate_settings', array( $this, 'add_filters_for_setting_validation' ) );
		add_action( 'customize_controls_enqueue_scripts', array( $this, 'customize_controls_enqueue_scripts' ) );
		$this->l10n = array(
			'invalid_value' => __( 'You must supply a title.', 'customize-validate-entitled-settings' ),
		);
	}

	/**
	 * Enqueue scripts.
	 *
	 * @action customize_controls_enqueue_scripts
	 */
	public function customize_controls_enqueue_scripts() {
		$handle = 'customize-validate-entitled-setting';
		$src = plugin_dir_url( __FILE__ ) . 'customize-validate-entitled-settings.js';
		$deps = array( 'customize-setting-validation' );
		wp_enqueue_script( $handle, $src, $deps );

		$exports = array(
			'l10n' => $this->l10n,
		);
		wp_scripts()->add_data( $handle, 'data', sprintf( 'var _customizeValidateEntitledSettingsExports = %s;', wp_json_encode( $exports ) ) );

	}

	/**
	 * Add filters for setting validation.
	 *
	 * @param WP_Customize_Manager $wp_customize Customize manager.
	 */
	public function add_filters_for_setting_validation( WP_Customize_Manager $wp_customize ) {
		/*
		 * Note that the priority needs to be greater than the default priority of 10
		 * that is set in WP_Customize_Setting::__construct() in the case that an
		 * existing filter has been added to sanitize the value, and so that we don't
		 * end up returning a WP_Error instance and cause any existing filters to
		 * erroneously manipulate a WP_Error object.
		 */
		$sanitize_priority = 20;

		/*
		 * Note that the following filters will only work for settings that rely on
		 * the `customize_sanitize_{$setting_id}` as is the default in
		 * `WP_Customize_Setting::sanitize()`. If, however, a setting subclass
		 * implements its own non-filter sanitization logic, then this filter may
		 * not apply.
		 */
		add_filter( 'customize_sanitize_blogname', array( $this, 'validate_title' ), $sanitize_priority );

		// Add filters to all nav menu items and widget instances.
		foreach ( $wp_customize->settings() as $setting ) {
			if ( $setting instanceof WP_Customize_Nav_Menu_Item_Setting || preg_match( '/^widget_.+\[/', $setting->id ) ) {
				add_filter( "customize_sanitize_{$setting->id}", array( $this, 'validate_array_containing_title' ) );
			}
		}
	}

	/**
	 * Validate a title field after sanitizing with ucfirst() for demonstration purposes.
	 *
	 * @param string $value Title.
	 * @return string|WP_Error String if valid, error if not.
	 */
	public function validate_title( $value ) {
		// Sanitize.
		if ( is_string( $value ) ) {
			$value = ucfirst( $value );
		}

		// Validate.
		if ( doing_action( 'customize_validate_settings' ) && '' === sanitize_text_field( $value ) ) {
			$value = new WP_Error( 'invalid_value', $this->l10n['invalid_value'] );
		}
		return $value;
	}

	/**
	 * Validate an array containing a title property.
	 *
	 * @param array $data {
	 *     Array containing a title.
	 *
	 *     @type string [$title] Title.
	 * }
	 *
	 * @return array|WP_Error Returns sanitized array if title is valid, WP_Error otherwise.
	 */
	public function validate_array_containing_title( $data ) {
		if ( ! array_key_exists( 'title', $data ) ) {
			return $data;
		}
		$validated_title = $this->validate_title( $data['title'] );
		if ( is_wp_error( $validated_title ) ) {
			return $validated_title;
		} else {
			$data['title'] = $validated_title;
		}
		return $data;
	}
}

add_action( 'plugins_loaded', array( 'Customize_Validate_Entitled_Settings', 'instantiate' ) );