A recent study summarized here (also reported here) says "no." The study finds that individuals who contribute more to churches also contribute more to other charities, and this result is robust to different ways of slicing up the data. The interpretation given is that churches and charities do not compete for money but instead help foster more spending in the other.
My first complaint is that this interpretation misuses the word "competition." Any dollar given to church rather than charity is, at the margin, a dollar not given to charity. There is an opportunity cost, which connotes a kind of competition. Saying that they do not compete is problematic.
My second concern is more substantive. Suppose a person decides to donate $X of her income to what she perceives as good causes. Then the decision of interest is how she should allocate those $X. She might give $X/2 to church and $X/2 to charity, $X/4 to church and $3X/4 to charity, and so on. Also suppose that there is diminishing returns to donating to more than one organization so each person wants to spread out donations, and that there is a cost, e.g., due to increased time writing checks or filing a tax return, to donating to too many organizations. Then the observed pattern is due to different people have different Xs, and not due to a donation to one causing a donation in another. In true, then there is a strict competition between organizations for a portion of people's donations due to a strong substitution effect.
However, other plausible complications can exist, and these are more subtle. Suppose churches teach their adherents to donate to non-church charities. If committed churchgoers, who happen to donate a lot to their churches because of their high commitment, also internalize the message to donate to non-church charities, then they could also donate more to charities than less committed types. We then see people who donate more to church also donate more to charity, but it is not the donations to one that cause the other. Rather, it is a third facto--in this case commitment--that causes high donations to both churches and charities. Sorting this out will require additional study.
What do you think? Does donating to church cause donations in charities? Vice versa? Neither?