Alright, I'll dispel the mystery. I'm talking about the LDS church. I always feel a little uneasy referring to it directly because I have friends and family who are members and being critical of it could easily be misinterpreted. I'll offer this disclaimer before proceeding: Some of my closest friends over the years have been members of the church and I can honestly say that they are wonderful people. However there is a faction of their membership that is so super-stinky self-righteous that it would take 100 members doing 100 good things to change my perception of the organization itself. They don't have those numbers so my negative perception persists.
I live at ground zero of the church and see the political moves that they engineer. Living here, it is impossible not to have a fairly intimate knowledge of what they're doing and of the public relations moves they make. I have also on occasion challenged members who I wasn't particularly afraid of offending and they've done nothing to correct any misperceptions I might have had.
The LDS church closely guards their numbers so estimates must be used in most cases. I base the numbers that I have used on a Newsweek article from about 20 years ago that estimated that they collect about 6 billion dollars a year in tithing. No one has ever challenged this figure. From a statement by the church itself, they proudly claimed that they had contributed 150 million dollars to charity over a period of I think 20 years. You do the math. It comes out to about 1%.
Since you seem to know something about accounting for religious institutions, maybe you can tell me if buildings and property are tax exempt. If so, that's where the egregious shortfall lies.
By "LDS church", I assume you mean the entire hierocracy thereof, rather than an individual church, yes?
Okay, let's start with the factual, information-based stuff first.
Buildings and property owned by a church are tax-exempt if their primary purpose is to be used for the tax-exempt activities of the church. Examples would be chapels, activities annexes, church-operated school facilities, parsonages, etc. If the church owns a building that is operated primarily for profitable reasons, then it becomes subject to taxes. Examples of this would be if someone left their house to the church, and the church chose to rent it out. There are, however, exceptions depending on what the rent money is used for (the IRS never misses a chance to make things complicated). If, for example, the house had a mortgage on it and the rents went to cover the mortgage payment, then different rules kick in. Also, if the rent money all goes toward charitable pursuits, there are different tax rules to cover that.
For the record, this no more constitutes an "egregious shortfall" than it does when the Red Cross blood donation facility is tax-exempt. (I use the Red Cross as an example a lot because I worked for them for a while, and can therefore provide a more in-depth, firsthand knowledge of what they do and how, FYI.)
Now, to edge into the more emotional issues you seem to be having.
I'm not surprised that the LDS church keeps their numbers private to the extent that the law allows. Virtually everyone does. Shockingly enough, religious institutions tend to have the same attitude toward financial information that individuals do: beyond a certain point, it's really none of your business. My church also does not publish its balance sheet for public consumption. However, this in no way implies that the information these institutions are required to provide is false, or that they're hiding something.
I can't speak to your Newsweek article, and without a source on the statement you mentioned by the LDS church, I can't really address that directly, either.
What I can tell you is that they appear to be about as straightforward and forthcoming about where they get their money and how they spend it as any other institution, so I can address that.
They are similar to my church in that they encourage the Biblical practice of tithing, ie. giving one-tenth of one's income to the church, and in designating that money primarily for operational concerns: building and maintenance costs, utilities, wages for paid employees, etc. Also in common with my church, they put some of that money toward missionary work, humanitarian aid, funding of certain areas of the universities they own and operate (although I will say I believe they have more of those than my church does). What seems to be unique to them is that they also fund the well-known family history program, and some of their tithe income goes toward that.
The LDS also have something they call "fast offerings", which my own church doesn't do, in which their members are encourage to fast for two meals the first Sunday of every month and donate the money that would have been spend on food for those meals to the church. Presumably, they're fairly diligent about this, because this provides a significant income stream by itself, which is distributed by the local churches to the needy in their own communities.
Back in common with pretty much every mainstream church I know of, they also take offerings and donations separate from these specific ones. That money either goes to specific funds earmarked for a specific, stated purpose - for example, if they hold a special fundraising effort for humanitarian relief for victims of a hurricane - or a general fund used for ongoing charitable efforts. Different churches tend to have different areas of focus, in keeping with the focus of their doctrine and their interests as determined by their members. The LDS church is very big on the concept of self-reliance, and many of their charitable programs have as their stated goal helping people to achieve self-reliance, and appear to include educational efforts as well as simply giving people stuff.
There is something that you need to remember: news articles on how much is spent and how generally skim over the surface of the topic and drastically oversimplify.
I have seen an article that was published in Business Week purporting that LDS charitable contributions equal about .7% of their income annually. Their information was drawn from a Welfare Services Fact Sheet put out by the LDS. The problem, as pointed out by the LDS in response to the story, is that the fact sheet only refers to certain of their humanitarian efforts overseas. It doesn't even touch on their domestic charitable expenditures.
Furthermore, there is a difference between money spent on charity and money donated to charity. The LDS church is different from my own - presumably, at least in part, because they are much, much bigger than mine - in that they not only operate their own charitable organizations and programs, they also partner with non-LDS institutions to provide funding. Therefore, when you talk about the church "giving money to charity", you are most likely talking about money they have given to outside charities, without addressing money spent on their own internal programs.
The LDS church, when it comes right down to it, operates under the same legal restrictions as any other church retaining tax-exempt status: they are required to put the bulk of their annual income toward their non-profit work. This is non-negotiable tax law in the United States, and whatever opinion one may or may not have about Mormons, I think it's safe to say that the IRS is very reliable in the sense that there's no way a church that large and visible would get away with cheating without having tax investigators up their asses with microscopes.