This can be solved using custom meta fields in the user profile (only visible to administrators) and then making an individual dashboard widget containing this meta (only visible if certain criteria is met).

Adding fields to the user profile page can be done:

1) manually

• Following Justin Tadlock’s tutorial: Adding and using custom user profile fields

• Using this Q&A to hide the fields for non-admin users

• Following Frank Bültge’s tutorial: Add WordPress Dashboard Widgets

2) by a plugin

The previous Answer used another plugin, but the Comments thread explains the change.
Appart from being well coded and maintained, Advanced Custom Fields is incredibly handy.

2a) Setting up Custom User Fields

^{click to enlarge}

2b) Administrator editing an user

_{/wp-admin/user-edit.php?user_id=7}

2c) Non-admin users viewing the Dashboard

The following will display (or not) each user a personalized message.

/**
 * Add Dashboard Widget
 */
add_action('wp_dashboard_setup', 'wpse_51591_wp_dashboard_setup');


/**
 * Only builds the Widget if the display message checkbox is enabled in the user editing screen 
 */
function wpse_51591_wp_dashboard_setup() 
{
    global $current_user;
    get_currentuserinfo();
    $show_msg = get_field( 'user_message_enable', 'user_' . $current_user->ID );
    $widget_title="Personal Messages to " . $current_user->data->display_name;
    if( $show_msg )
        wp_add_dashboard_widget( 'wpse_user_personal_message', $widget_title, 'wpse_51591_wp_dashboard_per_user' );
}

/**
 * Content of Dashboard Widget
 * Shows the content of each user's custom message
 */
function wpse_51591_wp_dashboard_per_user() 
{
    global $current_user;
    get_currentuserinfo();
    $the_msg = get_field( 'user_message_text', 'user_' . $current_user->ID );
    echo $the_msg;
}

Results in:

Bonus code
Adding a bit of UX into ACF: show/hide the Message Textbox if the Enable Message Checkbox is checked or not.
It’s better to change the order of the fields in ACF config, first checkbox then textbox.

add_action( 'admin_head-user-edit.php', 'wpse_51591_acf_profile' );
add_action( 'admin_head-user-new.php', 'wpse_51591_acf_profile' );

function wpse_51591_acf_profile() 
{
    ?>
    <script type="text/javascript">
    jQuery(document).ready( function($) 
    {       
        /* Wait 1.4s for ACF to be ready */
        setTimeout(function() {
            // find our checkbox
            var the_check = $('#acf-user_message_enable').find('input:last');
            
            // default state
            if( $(the_check).is(':checked') )
                $('#acf-user_message_text').fadeIn();
            else
                $('#acf-user_message_text').fadeOut();
                
            // live changes
            $(the_check).change(function () 
            {   
                if( $(this).is(':checked') ) 
                    $('#acf-user_message_text').fadeIn();
                else
                    $('#acf-user_message_text').fadeOut();
            });
        }, 1400);
        
        });
    </script>
    <?php
}