Think of conversation as a tennis match in which the players lob the ball back and forth. They ask personal or sensitive questions that put the other person on the defensive. If feeling inhibited is a problem for some people, others go to the opposite extreme: they use a date as an opportunity to purge and vent.
Again, when that point is will depend on you and what you think the other person is ready to respond to.
You may consider making a list and have them answer the ones they are able to answer.
What’s your biggest regret that you have in your life so far?
What’s the one movie you love more than any other movie? If you could only eat one food for the rest of your life, what would it be? What’s the funniest thing that’s ever happened to you?