I'm going to take another tack here. Someone mentioned "giving money to God" and this is the problem I have. If I send money to someone directly in need, is that giving money to God? Not according to the churches. The only thing that qualifies as "giving money to God" in the context of tithing is "giving money to the church."
Besides all this motive is the key. For an atheist to give to another it's not giving to God because he doesn't believe in God. He's simply giving for any variety of reasons and feeling good about it. When a Christian who loves and believes in God the motive and understanding is that he/she is giving it back to God who gave it to them in the first place.
40 And the King shall answer and say unto them, Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me. Matthew 25:40 (KJV)
Actually, I think you are both wrong. Giving to others with the spirit of charity is Godly, the scriptures don't say whether it is to a church, individual or really even define "needy". It also doens't qualify the giver or not, so yet, even when an atheist is giving in the spirit of charity, it is still "done unto me".
We talk a lot about why we give to who and when. More to the point, we talk about when we don't give and why. People say they won't give to the person on the street who asks for money because "they'll spend it on drugs or alcohol"... others say they won't give to a church because it's "just lining the pockets of the church leaders".
We are responsible for our own actions, not the actions of others. So, giving to either, in the spirit of charity is good for the atheist and believer alike, what the recipient does with the money doesn't change the charitiable act.
My wife and I give 10% of our income to The Church of Jesus Christ of Latter-Day Saints. There are other donations we make at the same time for specific charitable purpose. We also have our list of organizations we donate to which have nothing to do with church at all.
I know that the LDS church uses the money to build church houses, temples and other bulidings, take care of administrative expenses and other things. I also know that tithing money is used for non ecumenical purposes, such as sponsoring the BYU system, the polynesian center in Hawaii, which includes the pay and benefits packages for the faculty and administration. But he General Authorities of the LDS Church (who have been entrusted with the money) will be judged on how they used the money, not me.
By the same token, the Red Cross isn't a religious organization, but when I give to them, it is still in the spirit of charity, so it is still "unto me" (to quote Christ). Once in awhile there are instances when Red Cross officials or representatives have been caught defrauding the Red Cross. Does that change the spirit in which the donations were given? No.
I don't think it means we have to voluntarily be taken advantage of, and we were given judgement and discernment, so I also think we are expected to use them when it comes to donating... even in the spirit of charity.