We have made sure that the share feature in Coding Jam is completely safe. It's externally reviewed and vetted by the KidSAFE program: here's their Osmo Coding membership page.
No personal information from your child's account is ever shared (including your profile name). Along with the jam itself, the only thing that is shared is the randomly generated 3-word "band name". This cannot be tied to your account by the people who will see the song.
There is currently no option to disable the share feature.
