Hi, it's nice to hear from you!
You have some very good insight into this problem and that is going to help you. This is a common issue and it shows that your therapy is working. Transference occurs when a person feels safe with their therapist. They take deeper feelings- needs and wants- and they transfer them to the therapist. The therapist stays neutral in therapy for a reason- they need to allow a person to transfer issues into therapy. It is also so the client doesn't judge or use personal thoughts about the therapist to excuse, resist or otherwise hold back. For example, knowing that your therapist drinks alcohol might affect your desire to become sober. If the therapist stays neutral, however, then you are free to work on your alcoholism.
When you are transferring feelings of wanting to know your therapist's personal life, it is about something you need. It could be about control. Control can be about knowing a person's personal business. The more you know, the less threatened you feel emotionally. Or it could be about emotional neediness. That is not a negative thing. Being emotionally needy means that you did not get your needs met sometime in your life, most likely childhood. Your therapist is taking care of you. She cares and she listens. She helps you. These may be things you did not get as a child. So finding out about her, desiring to be her friend and be close to her is all about need. She fulfills a desire in you so you want more.
I understand your hesitancy in telling her about this issue. But you may want to reconsider. Transference is a very important part of therapy. It tells your therapist about needs you have that are hidden behind the normal exchanges. It touches a deeper meaning and need. By telling her, she can help you explore your needs and why you feel the way you do. It can take your recovery to a new level.
But if you chose to not tell her, then it may stay with you for a while. It may hinder your therapy. It will not harm you and it will probably go away after your therapy stops. But it is healthier if you tell her.
It may help if you tell her that you want to resolve how you feel. Let her know that you are aware it's not a good way to view her and you need to get to the bottom of it. If your therapist is a good one, she will see that she can help you.
Let me know if I can help any further,