Regarding your recruitment capabilities, a lot will have been determined on the day of your birth. Your nationality will often play a role, as well as your gender. I will divide the sections here in big nationality groups, most employers are restricted by. I will share here the recruitment possibilities I have personally come in contact with as Head of Mission or Core Team Member. This section covers only the recruitment possibilities for Short Term and Long Term Observers, not for Core Teams or Expert Missions which require a different set of skills.
Which nationality are you?
European Union (EU28 plus Canada, Switzerland and Norway)
If you have the nationality of one of the 28 member States or a State that has special agreement with the EU to send its citizens on its missions, you can apply to be a Short Term or Long Term observer in a European Union Election Observation Mission. Only citizens of these 28+3 countries can apply to EU missions. If you do not have the passport of any of these countries, do not even bother to register in the roster. Note that if citizens of Switzerland, Canada and Norway can apply as observers, they cannot be Core Team members. To my knowledge, and as of today, only Switzerland, Canada and Norway have a special agreement with the EU to send observers in missions. I will give some details at the end of this part on the recruitment procedures for nationals of these countries.
Official requirements to be selected as an observer can be found here: http://eeas.europa.eu/eueom/become-an-observer/recruitment/index_en.htm
Language requirements and previous country experience will increase your employability. Applying first to short term observers (STO) positions, in order to get specific experience is the best option, but if you think you have enough experience to be recruited directly as long term observer (LTO), you can try your luck. There is no official rule preventing your selection as LTO for candidates without STO experience.
To apply to EU missions, you should first register to the EU roster: https://webgate.ec.europa.eu/cas/. It is an online standardized CV form that will be used exclusively by EU recruiters to evaluate your profile. Fill it in with as much information as you can. At the end of the form you will have space for a motivation letter. I advise you to make it country specific and modify it every time, before applying for a mission.
Once registered in the roster, you have to apply through your ministry of foreign affairs, most of the time by contacting your national focal point for electoral observation mission, the procedure can be different depending on your country of origin. The EU does not recruit LTO and STO directly, your ministry of foreign affairs is the one selecting LTO and STO for EU missions. Focal points are employees of your national ministry of foreign affairs who pre-select candidates and propose their application to the EU. Out of the list of candidates pre-selected by the different ministries of foreign affairs, the EU will select a determined number of observers, depending on the size of the mission, balance of nationalities and other criteria internal to the EU. The list and contacts details of focal points by country can be found here: https://eeas.europa.eu/headquarters/headquarters-homepage/8775/eom-focal-points_en.
TIP: If you’re lucky enough to hold two EU nationalities, I would recommend you apply through the nationality of the country with the newest membership to the EU. For example of you’re French and Romanian national, apply as a Romanian. You will be in competition with less candidates if you apply with the nationality of a country that had less opportunities in the past to send observers to EU missions. France being one of the founder countries of the EU, it had the opportunity to send a lot of observers on missions and hence has more experienced observers than Romania that joined the EU in 2007, meaning you will face more competition as a French than a Romanian. The chances of having an experienced election observer with a professional command of French will be less if you apply as a Romanian. In short, apply with the nationality you think will give you the best chances to be recruited.
TIP 2: A good way to know when an European Union observation mission will be deployed in a determined country is to check for the advertised Core Team positions here: http://ec.europa.eu/dgs/fpi/announcements/jobs_en.htm. The field “Calls for candidates for EU Election Observation Missions and related trainings” is where Core Team positions are advertised. If job announcements are out for a Core Team, you can definitely send a message to your focal point to manifest your interest in an observer position.
OSCE/ODIHR (List of the 57 members States: http://www.osce.org/states)
The Office of Democracy and human Rights of the OSCE observes elections in the territories of its participating States. If you have the nationality of one of the OSCE country, you can apply to any of its election observation mission. Observers from the OSCE/ODIHR are recruited directly by members States. The institution does not maintain a roster, so there is no database to register to. In order to be selected, you should follow the recruitment procedure of your State of nationality. The procedure can usually be found on the website of your ministry of foreign affairs. (http://www.osce.org/odihr/elections/43585)
TIP 1: The focal point of your ministry of foreign affairs is often the same than for EU missions. You can therefore contact the same person if you want to apply for an observer position (http://eeas.europa.eu/eueom/about-us/focal-points/index_en.htm).
TIP 2: As for EU missions, a good way to know when a mission will be deployed is to monitor the advertised core team positions in the highlight section at the bottom of this website: http://www.osce.org/odihr. If you don’t see any ad for a Core Team position there, it means the OSCE is not recruiting at this time.
The Carter Center (All nationalities)
The Carter Center is one of the few, if not the only, international organization recruiting observers without restrictions of nationality. With relevant professional experience, country specific experience and language skills, you can be recruited by TCC. The trickiest part is to find the job offer. Here are two ways to find them:
Field-based positions, including for election observation (Core Team and LTOs) are now posted here (Thank you Stefan Krause for pointing at the update): http://electionstandards.cartercenter.org/about/opportunities/. From that page you can also sign up to their mailing list, as well as have access to preliminary statements and reports, which will increase your knowledge of their work.
The second option is to go to the website of Emory University, but it is mainly for employee positions based in Atlanta (http://www.hr.emory.edu/eu/careers/). Go to “Search for Staff Jobs (External Candidates)”, enter “Carter Center” and click search.