Hi Kate,
Sorry about your issues. It is not possible to use pictures from Flickr that are private in a Map Tour.
When you change a Flickr album from public to private, the pictures links will not be valid anymore and if you make it public account the picture links changes. This his how it should work for security concern but unfortunately it's poorly documented.
I think Picasa works the exact same way but you maybe interested by testing that.
Facebook works in a different way and it is possible to include private pictures from Facebook in a public Map Tour. Your other solution is to host them on a webserver, free solution like Dropbox works very well.