Can a coworker give out your personal info?

This is a policy question within each company.

If the coworker has access to you personal information through their job assignment, most companies would say that they cannot give it out, except under specific, limited, and well-defined circumstances.

If a coworker has some of your personal information because you shared it with them, or because they got it from a public source, they are under no obligation to protect it, unless that was an obligation you established when you shared with them.