You`ll need to pass “Oposiciones” exam before you can work in the public sector.
Here is a link to there site for info.
http://www.oposiciones.de/oposiciones_sanitaria/oposiciones_sanitaria.htm
There highly competitive, in high demand and very hard to get into.
Without it you cannot work in a Hospital or any other public sector job.
The conditions are very good, for a Spaniard it means a guaranteed job for life, you cannot be made redundant, the banks will lend you without much hassle and the hours and conditions are the best in Spain.
So many thousands apply for them.
You basically sit an exam, if you pass, you can then seek the job you require and start training. In many posts though they only employ people with the best grades, if not your given no choice on the jobs location.
A friend of hours was workin in a School in Malaga, she was placed in schools temprory to cover sickness etc.
Without much warning she was then told she had to take a temp position in Sevilla, which meant a 5 hour commute each day.
If she turned down the position she was goin to be expelled from the system.
So it has advantages and disadvantages.
I haven`t heard from any Expats who have passed them, I`m sure there are some.
Being from the EU you are allowed, but from outside the EU I don`t believe you qualify.
You`ll also need to show some educational qualifications for Spanish Language, you`ll need to have excellent language skills.