Why can I ask a question directed away from me, but as soon as conversation is turned to me I panic?

I guess you're hiding yourself behind walls all the time, maybe because you think you think you're not strong enought to protect yourself without a wall, or because you may believe there are ugly things inside of you that you're ashamed/scared of. Then this wall is the thing that prevents you from answering questions turned to you. Maybe shall you work upon what you're hidding behing your wall, and convince yourself that there is nothing to hide :)

because you may be frightened of people judging you and finding out information you feel is too personal for them to find out

Maybe because none of us may stand up to close inspection. It is easier to talk about the next person than it is about ourselves. We all want to be good and come across so too. We do not want to see our flip-side.

It sounds like you're uncomfortable with opening yourself up to others and that you don't want to seem vulnerable around other people. You don't have to share everything about yourself if you don't want to.

sometimes people do not feel safe enough to talk about themselves so they prefer avoiding talking about themselves

Some people feel uncomfortable talking about themselves, maybe for fear of being judged. Try observing whether you feel uncomfortable in every situation this occurs, or just in specific circumstances - you answer might lie there.

It can be really difficult at first to open up to someone about yourself. You may feel that you will be judged, or laughed at. But here on 7 cups everything is anonymous and confidential, you will get no judgement here.

As you may feel anxious that you have a large audience and you might have panic due to the fact that it's easier to entertain an audience if you don't know the audience exists.

We panic because we feel fearful, the thing we fear in this scenario may be the fear of humiliation or embarrassment. When the conversation is turned onto you you may think that what you want to say will lead to a negative reaction from those you are having a conversation with, for example they may laugh or criticize you.

I think for some people talking about ourselves can make us feel vulnerable to a point of panic. If it's a debilitating kind of panic, and you feel it's hindering your day to day life, it's important to find professional help. But if you find it is manageable, I think this is one of those situations where practice makes perfect. Ease yourself into it. Practice talking about yourself with someone you know you feel comfortable with. Talk out loud about yourself to yourself when you're alone. You shouldn't have to feel ashamed to share your life with others, but if it's hard for you, start small. Baby steps.