Facebook is the most complete database of people in the world, therefore it is the best for collecting statistics on Esperantists and promoting communication among Esperantists.
Facebook could be used as a census of Esperantists if a programmer creates a Facebook app that any Facebook user can activate and specify the level of proficiency in Esperanto: zero (nula), basic (baza), studying (studa), working (labora), fluent (libera). Of course, the app must be only in Esperanto to repell jokers who could lie about their level. Another mandatory feature of the app, so that it is not a one-time census but a permanent statistics on the number of Esperantists in the world, is self-cleaning. The app should annually ask its users to confirm information about them or to update it, warning that, if a user does not do anyting, he will be removed from the app. This will ensure reliable statistics that will not include dead people or those who is no longer interested in Esperanto.
This app will give us the following statistics: number of Esperantists in the world, number of Esperantists by level of proficiency in Esperanto.
Other features of the app could be a possibility to see the number of Esperantists in a city, to find out what city has the highest concentration of Esperantists, find Esperantists in a city, see the map of Esperantists, see what Esperantists are near you, find a date who speaks Esperanto, find hobbymates who speak Esperanto etc.
The creator of the app could collect donations from Esperantists.
:: Esperantist is a person who uses Esperanto.

I don't use Facebook (and don't recommend that people do), so I wouldn't donate for it, sorry!

I put "don't know", because I also don't use Facebook (unless it is required for something), so I wouldn't know if this would work or not.

It seems that there are about 505 Esperanto speakers on linkedin.com with Esperanto levels
Professional working proficiency
Full professional proficiency
Native or bilingual proficiency

I checked it using boolean Google search like
site:linkedin.com "esperanto=*=Professional working proficiency"
First Google shows much bigger number of search results but if you click the last page of results, their number will decrease.

Lernu.net has about 2300 Esperantists with "high" level of Esperanto.
Unfortunately there is no way to search people by language level on Facebook which is the most popular social network among Esperantists.

Well, there are a few problems with this:

1, Not all Esperantists use Facebook, obviously. If they do, it's perhaps likely that they're in the Esperanto group (15,281 members)
2. Even if an Esperantist is using FB, not everybody would be comfortable with the idea of being on some big list of Esperantists.
3. It would be difficult to get every Esperantist to voluntarily use this app.
4. Not every Esperantist considers themselves part of "the movement". (I'm not even sure what qualifies one as part of "the movement", but some of my friends don't want to be associated with it)

Also in my opinion, let's not be gatekeepers, based on someone's experience level; "only those who are Professional Working Proficiency+ count". Perhaps just taking an interest in Esperanto is enough to make one an Esperantist.

There are probably better ways to estimate the size of the Esperantist demographic, and it may use software/technology to help achieve that, but I'm not very experienced with doing that, myself.

I'm not sure if you'd want donations to hire a programmer for this project, but I don't think this would be of a scale to need to hire somebody. (The app idea seems relatively simple). Some of us write code for free.

I agree with Moosader here. Many, MANY people who might be competent Esperanto users do not have anything to do with Facebook, Google+, Twitter, or whatever others of the so-called "social media." (For example, I for one will not touch them, regardless of my level of use of Esperanto.) Therefore, trying to judge the E-o community by something such as Facebook (or whatever) could be almost wildly misleading and inaccurate. I consider this proposal to be a serious loser.

I would be happy to give a small donation to any Esperanto cause but only if it is easy to do so. I don't want to use my credit card online. Is there was an easier and safer way to transfer money, I would probably give about 20 dollars.