Yes, they definitely need to make it both public and private. I've done both and enjoyed the public far more given the many varieties of the different builds being played. As someone mentioned, the private squad can be a bit more picky, thus eliminating a lot of people who have just joined or returned to GW2 and aren't totally proficient with their skills. To be fair across the board, the Marionette fight needs to be open to both on a permanent basis.