Not where I live. Not only to you have to be a member of the Catholic church to go to the Catholic schools here, they ask that you tithe 10% of your income. If you cannot afford that, you have to prove that you cannot. But you still have to belong to the parish.
You have to belong to the parish most of the time, but that can be relaxed for a number of reasons. Around here, you have to state which parish you belong to, because not every parish has a school. They do request that you donate what is subsidized by the parish, so that they can recoup their expeditures as much as possible. However, it is just as commonly requested that you be a regularly donating member (weekly, monthly, bimonthly, etc.) or to offer your time and talents volunteering at the school or parish. There is also usually a "non-participating" tuition rate that is higher for those who aren't Catholic, but for some schools, there are long waiting lists. Usually they are the ones that have better success rates, and can "afford" to be a little picky, because there is always another student waiting to get in to replace the one who left.
Like I said earlier, I've had my kids enrolled both ways and I don't prefer one over the other. I look for good schools regardless of whether public or private. Neither are perfect, and neither should have a monopoly.