Your intent as the original owner of the data is important! You have to explicitly give Linkedin the right to display your data. It's in their Terms of Service. If Linkedin does something with your data that is outside the ToS, then that is on them, but if they do something within the ToS that you don't like then maybe you should not have provided them with your data.

As for whether the data should be public, that's a decision we each have to make.