to tutor has been formed from the noun a tutor, mainly because there wasn't a verb for it before that, so saolandamy is (or are) correct, that's the closest you'll get.
Actually, as long ago as the 19th century, tute has been used as a joking, colloquial abbreviation for the verb or noun tutor. So, if you mean to use the word in a very informal setting, why not use it?
Someone who is being tutored is a "tutee". "Tutelage" also comes from the same family of words; from the Latin "tutela" meaning "protection" or "guardian", which in turn comes from "tueri" meaning to "look at" or "guard". In any case, a decent etymological dictionary always comes in handy for these kinds of queries.