I got the same error in 3.8. But it can be fixed. (Please note, I am just a programmer, not the author of the plugin, and no warranties are expressed or implied.)
The problem seems to be that WordPress, somewhere along the line, changed the function prepare() to take two arguments instead of only one.
Open the file shortcodes.php, located in the wp-content\plugins\multisite-global-search\inc directory, in a text editor.
Go to line 223. (If you don’t have a text editor with this functionality just search for “$request = $wpdb->prepare”.) Replace this:
$request = $wpdb->prepare( "SELECT ".$wpdb->base_prefix."v_posts.* from ".$wpdb->base_prefix."v_posts left join ".$wpdb->users." on ".$wpdb->users.".ID=".$wpdb->base_prefix."v_posts.post_author ".
"where ".$wheresearch." ".$termsearch.
"AND ( post_status = 'publish' OR post_status = 'private' ) AND ".$post_type." ORDER BY ".$wpdb->base_prefix."v_posts.blog_id ASC, ".$wpdb->base_prefix."v_posts.post_date DESC, ".$wpdb->base_prefix."v_posts.comment_count DESC" );
with this:
$request = $wpdb->prepare( "SELECT ".$wpdb->base_prefix."v_posts.* from ".$wpdb->base_prefix."v_posts left join ".$wpdb->users." on ".$wpdb->users.".ID=".$wpdb->base_prefix."v_posts.post_author ".
"where ".$wheresearch." ".$termsearch.
"AND ( post_status = 'publish' OR post_status = 'private' ) AND ".$post_type." ORDER BY ".$wpdb->base_prefix."v_posts.blog_id ASC, ".$wpdb->base_prefix."v_posts.post_date DESC, ".$wpdb->base_prefix."v_posts.comment_count DESC", "" );
Now go to line 225. Replace this:
$request = $wpdb->prepare( "SELECT ".$wpdb->base_prefix."v_posts.* from ".$wpdb->base_prefix."v_posts left join ".$wpdb->users." on ".$wpdb->users.".ID=".$wpdb->base_prefix."v_posts.post_author ".
"where ".$wheresearch." ".$termsearch.
"AND post_status = 'publish' AND ".$post_type." ORDER BY ".$wpdb->base_prefix."v_posts.blog_id ASC, ".$wpdb->base_prefix."v_posts.post_date DESC, ".$wpdb->base_prefix."v_posts.comment_count DESC" );
with this:
$request = $wpdb->prepare( "SELECT ".$wpdb->base_prefix."v_posts.* from ".$wpdb->base_prefix."v_posts left join ".$wpdb->users." on ".$wpdb->users.".ID=".$wpdb->base_prefix."v_posts.post_author ".
"where ".$wheresearch." ".$termsearch.
"AND post_status = 'publish' AND ".$post_type." ORDER BY ".$wpdb->base_prefix."v_posts.blog_id ASC, ".$wpdb->base_prefix."v_posts.post_date DESC, ".$wpdb->base_prefix."v_posts.comment_count DESC", "" );
This all looks worse than it is. You’re just adding a second empty argument to the wpdb->prepare call so it can be recognized.
If you do this correctly your error will go away.