Religion... shall mean for us the feelings, acts, and experiences of individual men in their solitude; so far as they apprehend themselves to stand in relation to whatever they may consider the divine.

James might have meant something different by emphasizing solitude than what you take him to task for. He continues:

Since the relation may be either moral, physical, or ritual, it is evident that out of religion in the sense in which we take it, theologies, philosophies, and ecclesiastical organizations may secondarily grow. In these lectures, however, as I have already said, the immediate personal experiences will amply fill our time, and we shall hardly consider theology or ecclesiasticism at all.

The surrounding context is that of delineating the overall theme of the book. James wants to focus on the religious experience rather than the religious convention: church rituals, theological debates, Sunday school. Leaving the social institutions to sociologists, James pursues the psychological experience, including that raw feeling of sacredness you mention. I don't believe he insists on loneliness (Frank may be - I haven't read Frank), nor that loneliness is in any way important to him.

You ask:

Is the feeling private in the same sense that we have difficulty communicating any experience? Then why emphasize this of sacredness, rather than sneezing?

I think it may be because there aren't churches devoted to sneezing in special ways, dogmas of righteous sneezers, rituals of mass-sneezing and so on. It's possible to talk to someone about their sneezing problem (or, alternatively, their sneezing as a solution to many problems) without them parroting conventional truths about sneezing that they have internalized.

The solitude James speaks of is simply that of the internal dialogue with yourself. You may experience it together with many other people, indeed even standing in a crowd with them, and it need not be unique, but so long as you are alone with it in the confines of your mind, it is personal. Not unique, not even necessarily inexpressible - if James thought it were, his would have been a one-page book - but at its origin, intensely personal. Yours to experience, interpret, act upon and try to communicate as you wish. Yours alone. Yours - alone.