wp_get_archives( ) for specific author?

You can add filters to wp_get_archives().
Call wp_get_archives() like this for logged in users:

<?php
 	if ( is_user_logged_in()) {
		global $archive_author_id;
		$archive_author_id = get_current_user_id();

		// add the filter to wp_get_archives to filter it by user
		add_filter( 'getarchives_where' , 'getarchives_where_filter');

		wp_get_archives('type=monthly'); 

		// remove the filter
		remove_filter( 'getarchives_where' , 'getarchives_where_filter');

	} else {
		// user is not logged in
		wp_get_archives('type=monthly');
		echo 'You have to log in to see your personal archive';
	}
?>

And put this in your theme’s functions.php:

// alter the sql "where" query for montly archives
function getarchives_where_filter( $where) {
	global $archive_author_id;
	return $where .' AND post_author = ' . $archive_author_id;
}

// query monthly archives by user.
function my_post_queries( $query ) {
	// do not alter the query on wp-admin pages and only alter it if it's the main query
	if (!is_admin() && $query->is_main_query()){

		// alter the query for monthly archive pages
		if(is_archive() && is_month()){
			if ( is_user_logged_in()) {
			  $user_nicename = get_userdata(get_current_user_id())->user_nicename;
				$query->set('author_name', $user_nicename);
				//$query->set('posts_per_page', 1);
			}
		}
	}
}

add_action( 'pre_get_posts', 'my_post_queries' );