Third party add-on's data does not get duplicated when you duplicate current channel entries into a new MSM site. EE also doesn't have hooks for this, so it isn't easy for any third party add-on to automate this at all. As far as I know, very few or none even provide a tool for accomplishing this.
You'll need to write a script to port data from one site to another. You'll need to take a look at the database tables Channel Images uses and work out how to duplicate your data associations from your old
site_id, to your new ones (the new, duplicated channel entries).
I've done this with other add-ons but not Channel Images. You'll have to work it out yourself; you may want to contact the Channel Images author if you run in to trouble understanding their data structures.