I'm considering a private religious school as well. I'm guessing that you are talking about a Jesuit school? You probably know that they are very into public service. If you think that is the type of law you might be interested in, they would probably be a very good fit for you.
Anyway, to answer your question, I have heard that there is very little, if any, inclusion of overt religion in those schools. The "religious" aspect will come across more in the values of the school. As I said above, they will focus much more on public service and doing right by your fellow man, as opposed to the cut-throat law that some other schools push.
As far as being an alum of a religious law school, I understand that it can help you out. If you are interviewing for a job where a religious background could be a positive (or if you know the people interviewing you are very religious), then you can sort of put it out there that you went to a religious school, which will imply that you value religion. This wouldn't come up often, if at all, but it can be a useful "card" to play.