Even though you can benefit from doing a search of other threads on this board specifically answering this question, I will tell you that No (rycal) you specifically do not.
This is not to say that some people will not benefit from taking some sort of teacher training such as the TEFL certificate or CELTA certificate gives you.
If you have no clue or are inexperienced TEFL / CELTA or other teacher training can help to inform and prepare you so that you can decide if you want to even do the teaching thing or not and either way help prepare you to give better thought out and organized lessons that can make actually doing it easier and more enjoyable.
Also. In other less desirable places to be than Spain it can be required to get some of the better teaching jobs making it worthwhile.
Now for everyone out there thinking about coming to Spain to do the teaching thing to survive (usually because there are no other real job options for native English speakers especially the ones living here illegaly).
Quote, frame, highlight, and dig on the following:
No. You will not need TEFL / CELTA etc. if you plan to only live and work in Madrid.
If you have experience, courage or even just a clue as to how to tutor or teach and are coming to Madrid to see what happes and live an adventure then you do not need any kind of TEFL / CELTA or other type of special training or certification to find English teaching or babysitting / aupair type jobs in Madrid.
You will be able to find both private lessons through the local papers and notice boards and you will even be hiried illegaly or not by some academies who will send you out to businesses and such to teach English.
The pay is crap 9 euros to 18 euros and hour but you will be in Spain. You would be sacrificing the higher pay of all the other less desirable places in order to be in this more desirable one.
The English teaching job market has its ups and downs here but you can just keep moving forward replacing your bad hours with better ones and keep adding on more new ones as you roll along.
At times it will seem as if the jobs are not coming in and that you do not have enough offers / hours at any given time, but with patience and persistance and constantly reminding people that you are here and available for work you will survive well enough.