The Church will never say "no" to someone who wants to go on a mission and can't afford it.
I think this is especially true for foreign missionaries. Can that missionary from the Dominican Republic really afford $500/month? Doubt it.
Nutshell - I know that you already know most, if not all, of this - but it's for the benefit of others
There is a lot of variance in how much a mission "costs". Pretty much every missionary from the US (and probably from Europe) pays about $400 per month (or thereabouts). I believe the money goes into a general fund that is then used for expenses on the mission. Every month when I was in Peru we went to the bank to withdraw our money. I'll do the numbers in "Soles" (the currency of Peru) because that is what I know. $1 = 3.5 Soles (approximately). We usually paid somewhere between 150 and 200 soles for rent, and we paid about 300 - 400 soles for "pension" (basically the "board" part of "room and board". In my mission we never cooked for ourselves, we gave someone the money at the beginning of the month and they bought and cooked our food. Usually they were able to cover most of the food for their entire families off of what we gave them). About 20 soles for bottled water. Then we got about 100 soles for personal use. So, if we add that up: I consumed about 450 - 500 soles (only half of the rent, pension and water would be allocated to me - the other half to my companion). Every so often my mom or grandma would send me a $20 bill in a letter (not a good idea to send cash through the mail to Peru, but I think I got most if not all of it). I was never hurting much for money.
So, I put in $400 = 1400 Soles, but I consumed about 500 Soles. The other 900 soles went to helping out some of the Peruvian missionaries who couldn't afford even the 500 soles per month (and, in general, much less was required of any Peruvian missionary). Some of it also probably went to subsidize missionaries in places like Europe where a typical missionary consumes more than $400 per month.
The church handles and distributes the money for missionaries, but it is money that has been donated specifically for that purpose (mostly by the parents of the missionaries themselves). That is mostly because you don't want the missionaries to be having to deal with large amounts of money at any one time. It's bad enough being in some neighborhoods wearing suits, but at least most of the muggers know that the missionaries don't have much that's worth stealing.
Someone mentioned before about missionaries looking like they need cars. That is entirely dependent on the area they are in. Some areas are "car areas", others "bike areas". It isn't a matter of whether or not the missionary can afford a car. Those who are in car areas also have pretty heavy restrictions on how many miles they can use in a month too, so they often try and find members in the area that can give them rides whenever possible. There were no "car areas" or "bike areas" in my mission. We either walked or forked out a few pennies to take a bus. (I lost about 60 lbs on my mission
It's all back though).