It varies greatly depending on the locale.
In my area, most SAR is volunteer based. Technically the Sherrif is in charge. Usually the sherrif's department sends one officer to stay in the command center and offically manage the search. Typically Emergency Management will also send a person or two. These positions are typically 8-12 hr shifts and the actual person changes but the agency is continually represented. Typically local police and fire department are represented. The bulk of seraching is usually performed by either volunteer teams and/or fire departments (of which most are volunteer).
I am a member of a SAR team which uses both Dogs and Man Trackers. Man trackers and Dog teams search in similar but different manners and are used differently in the field.
Out west SAR teams are often part of the local Sherrif's department. Horses can be employed due to the large areas to be covered.
I would recommend calling either the Sherrif's office or Emergency Management about teams in your area. They should be able to tell you who performs search and how to contact them.
We do not require any previous experience to join our team. We obviously would prefer people with outdoor experience, moderate physcial stamina, high availablity, first aid skills, and a desire to serve. The dog portion of the team trains at least 3 times a week year round. The ground searchers train one weekend every other month specifically on search. But we are also a technical rescue team so alternating months we are training on swiftwater, trench collapse, building collapse, HAZ MAT, and rope rescue.
Most teams require you to earn a CPR certification and at least basic first aid before they will allow you to deploy. NASAR offers a variety of classes and certifications specifically about search skills. Most teams always need more volunteers and run in house training.
Most teams have very little money and equpiment, fuel, travel, and all other expenses are covered out of pocket by the volunteers. Most teams use pagers to alert members of a search. For some reason we normally get paged around midnight to 2am for searches and deploy via personal vehicles directly to the command center of the search.
If you don't mind tromping around in the dark in unfamilar woods in foul weather with very little chance of ever being recognized for your effort, then Search is for you.