Teachers are employed by a school district. In the Southeastern US, these school districts represent the county population, and there is often one school district per county complete with a superintendent, board members, and many heads of various departments at the county office level---such as: the head of curriculum development, the head of finance and payroll, human resources directors and assistant directors, the director of transportation, foreign language and gifted department heads, science and mathematics department heads, English and ESOL department heads, reading specialists, PR people who represent the school district, as well as whole legal departments in some larger, wealthier counties. These "heads" earn the lion's share of the monies that come into the school district and their pensions are quite fat because they earn so much. Teachers are actually the "low men" on the pay scale, just a few rungs above paraprofessionals and teacher support specialists like librarian aides and bus drivers. There are also several staff members at each school within the school district whose functions are to keep the everyday business of the school running smoothly such as: various clerks, accountants, discipline and attendance people, etc. Their pay is often similar in scale to the same type of jobs in the private sector. None are Federal employees. They may or may not have a defined benefit plan depending on when they were hired. None of these employees, including teachers, ever have to compete with any other business for "sales" since public school attendance is mandatory to a certain age, but all employees in a public school/district capacity must mmet state, county, and often federal guidelines to keep their jobs and they often must compete for jobs within the district/state. Educational level/competence and one's level of professionalism is your ticket to ride, not how many widgets you can sell. So it's a different world from the corporate one. It also helps to "know someone" if you want a job in education in the South.
Some school districts also employ school psychologists who often serve several schools, psychometrists, school social workers, physical therapists, art teachers, band and choral directors, industrial arts teachers and "home arts" teachers. These people are also NOT federal employees. There are "registered nurses" in some school districts in the South, and most counties employ at least one dietitian. (And of course, there are "the lunchroom ladies" who make our delicious hot breakfasts and lunches daily.) These people are also NOT federal employees.
Teachers' pensions are often moderate and fairly ho hum compared to those "heads of departments" who have offices at the county district office, no matter what you hear in the news. Every once in a while a teacher may make news headlines because she earned many, many higher level degrees (one way to increase your teacher pay on the pay scale ladder) and will work 40 years or longer (another way to increase your teacher pay) so that her salary and eventual pension appears to be outlandish, but this is not the norm for the majority of teachers. If you are ever concerned about outlandish salaries and pensions of education personnel, you should trim the "fat" at the top.
Teachers in the SE area of our country are most often paid from monies from the state, a little bit may come from Federal monies in the case of say, some special education teachers, or teachers hired for a particular focus that happens to be "the darling" of whoever is president at the time (for example President Obama was in favor of all students having equal access to the best/same education/curriculum available, not just students in wealthy counties, and he was instrumental in helping to move STEM educational efforts forward. He backed up his word with "Race to the Top" federal monies that states could earn if they met certain Federal guidelines. Teacher preparation, curriculum development, and teacher training/evaluation was tied to this money, so teachers were affected in positive and negative ways by "Race to the Top" --federal dollars--depending on your point of view.) The federal government often uses money/funds promised to states as a way to change the way education works or to change the focus of education. The Feds will let states have that money as long as they do what the feds want. If you don't like Federal interference in local school decisions, you probably would not support a program like "Race to the Top". If your school system is well off and you employ the best educators and have the best resources, you probably do not want the Feds in Washington telling you what you can and cannot do in your teachers' classes, and you probably have little incentive to take Federal monies unless the whole state votes/makes your county do it. But if your county is poor and you have few supplies for your students and your teachers are near exhaustion from the daily uphill battle against crime, drug use, poverty, ignorance, apathy, etc in your county, you might view help from Washington in the form of mandates, laws, and especially money, more favorably. So even though teachers' daily lives, and their students', can be influenced by the Feds, teachers are hired by local school systems and even by a local school principal in some cases. Most teachers attended local universities in their area of the country and most teachers actually know their principal before they are hired because she either goes to the principal's church, knows a friend or a fellow teacher who teaches with the prinipal, has coached or golfed with him or the principal's spouce, their kids attend the same camps, etc. This is often how you are hired as a teacher in the South. But no, teachers and principals are NOT Federal employees. The majority of a teacher's salary, and in many cases, all of her salary, comes from state and local funds. Most states now have (in 2015 - 2016) an open records website where you can view each teacher's, principal's, and even college professors' salaries. You can even view the governor's salary....just in case you are curious.
And just a word about teachers' unions---in the South, many teachers do "join" a teachers union of some sort in order to either get some sort of necessary insurance, get additional education/training and support/sense of camaraderie, or to support efforts by their union to get and keep teachers' benefits and to increase teachers' salaries. But at the same time, all teachers in the South know that our unions face an uphill battle and basically are not "real unions" in the sense that those strong unions of the Northeast used to be. Here in the South, our teachers' unions are viewed more as a club than a union, and often, sadly, teachers' unions here have no real power. It's only the election of a governor who is sympathetic to the plight of teachers (one whose mother or wife is/was a teacher perhaps) and high times on the economic front that determine whether or not teachers will see pay and benefit increases, and yes, these mainly come from the state and local governments. In good economic times, no one questions the, often low, salary a teacher earns. But when times get hard, every citizen suddenly decides the government cannot afford to pay teachers, policemen and firemen much of anything. Suddenly the green monster emerges. If you are a teacher, keep reminding everyone you know of what you do and just how little you've earned over the years doing it. For example, I just retired after about 35 yrs of teaching "on the floor". When I began teaching in 1979, and for most of my first 10 years of teaching, I earned less than 15,000 a year. (My first year I earned less than 7000.00.). All my friends earned two and three times what I did, in the private sector, and many invested their money and eared over 10% at the time, which "they rubbed our noses in" monthly. I never had enough money left over to invest during those good times. We teachers just kept plugging away at our jobs, which most of us loved or we would never have stayed under such conditions. Fast forward to 2008. I was soon to retire with a pension after 35 years of excellent service for average and even below average pay. And you ought to hear all the belly-aching from some of my friends and relatives who have lost more than they care to talk about in the recent downfall. Now, for some reason, they think we teachers are the new barons of Wall Street. Whaat?? Get real. Look at the overall facts. You cannot change the rules because you spun the wheel and don't like your odds in your later years. But even so, no, we are NOT federal employees.