Non-profit and educational institutions tend to only use outside recruiters for very high level positions because they’re very expensive, charging anywhere between 10 and 25% of a year’s salary for a permanent placement. Such organizations very rarely want to spend that kind of money, especially when they usually have more applicants than they know what to do with. However, if you have very specialized skills that are in high demand, someone may seek you out.
I’ve never known a recruiter that started with a candidate and then went and sought out possible employers, usually because candidates can’t afford the fees. They may keep you on file until a possible opportunity comes in, but that’s a passive process.
I kind of have a distrust of recruiters because their loyalties are torn between two masters, candidate and client, but only one master pays the rent. Some will push hard to close a deal that may or may not be in your best interest.
It’s better to find professional organizations in the fields and geographic areas in which you wish to work. Often those groups will have job boards that you can search, and you may meet other people already working at a possible employer that can give you insight into how things work there.
Good luck and be careful out there.