Prevent users in the backend from seeing WP/Plugin notifications and update annoucements?

Add this to your functions.php.

if ( !current_user_can( 'edit_users' ) ) { //Change the edit_user" to whatever capability you need to retain the notifications
  add_action( 'init', create_function( '$a', "remove_action( 'init', 'wp_version_check' );" ), 2 );
  add_filter( 'pre_option_update_core', create_function( '$a', "return null;" ) );
}

Or if you want to show the notification to a specific user only you may try something like this:

global $user_login;
   get_currentuserinfo();
   if ($user_login !== "admin") { // change admin to the username that retains the notifications
    add_action( 'init', create_function( '$a', "remove_action( 'init', 'wp_version_check' );" ), 2 );
    add_filter( 'pre_option_update_core', create_function( '$a', "return null;" ) );
   }