If one has $100 to give to charity, is it better to give it all to one person, or distribute $1 to 100 people (so that if one isn't legit, the others will cover)?
The Chofetz Chaim, Rabbi Yisrael Meyer HaCohen Kagan, says that this explains how we should give Tzedakah, charity. He says it is better for an individual if he or she gives 100 individual dollars to 100 poor people, than all to one person. This way, one becomes accustomed to fight against his miserly inclinations 100 times, and it becomes easier to win in the future. Furthermore, giving Tzedakah becomes a completely natural act, given that one has become accustomed to giving by doing so such a large number of times.
I just found this story about a Chassid who, when asked for charity, would always give it in two parts, so that he could fight his inclination twice for every time he gave charity.
I'm less concerned about "if one isn't legit"; a better argument is that you've done the mitzva of tzedaka one-hundred times, which may have more of a positive effect on you than one large donation.
Usually the overhead is such that recipients do far better with a few larger donations; on the other hand, there may be numerous causes, and the argument above. So I think it all depends on the situation and the person.
The story is told of a fellow going off to WWII; he asked Rabbi Naftali Riff of Camden, NJ for a bracha. Rabbi Riff asked him for a $100 donation. What Rabbi Riff then did was send a $10 donation to a different organization every six weeks or so, so the soldier got a thank-you letter every so often from different Jewish organizations! That helped keep his spirits up.
I understand that you're coming from the standpoint of meshullachim in shul who come asking for whatever you can give, or also someone begging on the street for whatever he can get.
When you have your option of charity organizations (local or far away) to give to, then legitimacy usually isn't a concern.
When someone comes up to you with a specific request, then you can usually find a particular dollar amount that will fill that person's need. You should try to fulfill as many people's needs as possible. If someone comes to you asking for $100, and you have $100 to give, then you shouldn't decide instead to give $1 to each of 100 people. The person who asked for $100 may not find someone else to give him the remaining $99.