What information can people access on my lemmy user page?
My own page shows my subscriptions but it seems that's just visible for me. Can other users see my subscriptions? Can I turn that off?
If I look at another user's page there is a tab "saved". Does this mean other people can see what posts I save? Can I turn that off?
We can only see what you posted and commented. The saved button is going to be removed in the next update
So we won't be able to save posts? Or just the button will be hidden, but still menu-accesible?
It will just be shown if you're on your profile page :) (i.e. where it is actually useful)
Note that the admins of your instance can see absolutely everything in your account. Social media is not private and Lemmy is no exception to this.