Use the checked function.
https://codex.wordpress.org/Function_Reference/checked
The HTML you want to render should look like this:
<input ... value="hide" checked="checked" />Hide from these locations.</br>
<input ... value="show" />Only show to these locations.</br>
What you’re getting instead is probably somthing like this:
<input ... value="hide" />Hide from these locations.</br>
<input ... value="" />Only show to these locations.</br>
This should help:
<input type="radio" id="edit-menu-item-visibility-<?php echo $item_id;>" name="menu-item-show-hide[<?php echo $item_id; ?>]" value="hide" <?php checked( get_post_meta( $item_id, 'hide_show', true ), 'hide', true ); ?>" />Hide from these locations.</br>
<input type="radio"id="edit-menu-item-visibility-<?php echo $item_id; ?>" name="menu-item-show-hide[<?php echo $item_id; ?>]" value="show" <?php checked( get_post_meta( $item_id, 'hide_show', true ), 'show', true ); ?>" />
Only show to these locations.