Many Auxiliares in Madrid teach private language classes to insulate their paycheck and earn travel funds. People generally charge between 10-20€ per hour but you should look around on the Auxiliares private classes pages to make sure you are asking a fair rate.
Some good places to find private students and people interested in conversation exchange are Tus Clases Particulares and funnily enough, Couchsurfing.com, which has a big network of groups for interests in addition to language learning such as football and rock-climbin.
Another program is called Mingles, and they host language exchanges in bars. I know some people who have worked for them although not long term. They hire people to lead the discussions and though you do get free beers, your hourly rate isn’t all that high. A euro is a euro and a beer is a beer though!