I work in Tenerife as an Auxiliar Inglés and have been teaching for 5 years now. I went back to the UK to do my CELTA just over a year ago. If you want to teach in Tenerife as an English Teacher in State School you will need a Degree that is then translated and approved by the Canarian Government. A PGCE would be good too. You will then have to do your oposiciones and you could be given a position anywhere in Spain until you make your way up the 'list' and get enough points to request a school where you wish to be.
It's a complicated system... by far too complicated to write the whole thing here!
You could become an auxiliar, like myself, which means that you work in a school alongside the English teacher. Due to Spanish law, you will not be left in the classroom alone and are not classed as the 'teacher'. Although you do not specifically need a TEFL for this work, it's preferred and the waiting list is long for jobs, so if you have one it would help. I would get a CELTA if I were you.
For private schools, I would assume at least a degree is required, a PGCE and a CELTA too, but I cannot speak for individual schools. You would have to contact them directly and ask what their requirements were.
Hope this helps :)
For more specific info about Tenerife, you could try www.tenerifeforum.com
as there is loads of stuff about the Spanish (and specifically Tenerife) education system. I have written a lengthy post about it explaining all about the system and it's trials and tribulations! Just do a search for my username, as it is the same on there, as it is on here.