I'm wanting to use a saved search as a parameter on a M/R script. But the only way I can get it to show in the list of saved searches it to make the search Public and available to everyone. Is there a way around this?

If your parameter is a list of saved searches then as far as I know no. But if you change your parameter to free form text, you can put in the saved search id. Then in your saved search, under Audience tab you can set it to the audience you want.