Expecting a religious person to keep their religion private is much the same as telling someone that they can fall in love but must keep it a secret. It's a waste of time. Anyone seized by love will be quite unable to do what you ask. You'll see it in their eyes.
But that's no justification to bare the whole of the relationship to the world. Imagine a couple in love who broadcast their entire life to the world. All the whispers, the kisses, the lovemaking, the arguments, the drudgery of housekeeping... Some things are best nourished in private, and the same is true with religion.
Religion must have public and private aspects. There are parts of religion which are wholly internal just as much as there are parts which affect the public life of the believer. It's unrealistic and inauthentic to confine it in total to one or the other.