Matthew 6:6 English Standard Version (ESV) But when you pray, go into your room and shut the door and pray to your Father who is in secret. And your Father who sees in secret will reward you.
A primary truth that is required but does not affect our eternal salvation is prayer. Prayer in this sense is meant to be a personal conversation. So often we assume the position of talking and spend very little time listening. What kind of relationship do you have with anyone if only one party speaks and when they are through, they get up and leave? This is about relationship and what you make of your prayer time sets the tenor of the relationship.
The issue of privacy is one of not allowing distractions to interrupt the conversation. Most of us see anything that makes noise is a distraction, like family, television, radio, traffic and such. What if the greatest distraction is your inability to calm your fears, doubts, pains and worrisome troubles? You will often see the necessity for prayer to be any or all of those internal distractions. How can you listen for that still small voice if you cannot quiet your mind after you have talked about them?
At this point I could give you a list of things I do or do not do during my quiet time. This is your relationship and not mine. God wants a relationship not a set of rules to be followed. Any formula for success in any relationship has to be sincerity. Be honest with yourself and develop a relationship with God that is personal.
Just as an added note because it is important, that secret place is in your heart.
Ephesians 2:22 English Standard Version (ESV) In him you also are being built together into a dwelling place for God by (in) the Spirit.