The four sunni schools are not by any means to be taken as some kind of sectarianism, rather there has been great harmony amongst the scholars of the 4 schools throughout islam's history.
Imam Abu Hanifa was the 'founder' of the hanafi school living not long after the prophet's death in modern-day iraq. Imam Malik came shortly after living in the city of medina. There are some reports that they did live at the same time, although Malik was much younger, and their mutual respect is well-known. In fact, one of Abu Hanifa's main students from whom a lot of the hanafi school relies on, studied from Imam Malik aswell.
Imam Shafi'i was also taught by both Abu Hanifa's students and Imam Malik and his respect for both men is also well-documented. Imam Ahmad ibn Hanbal studied with Imam Shafi'i and consequently there are many similarities between the maddhabs.
It is important to know that all 4 schools have the correct guidance according to the sunni's and the differences lie not in the fundamentals of faith rather on finer judgements and jurisprudence, and are a result of the independent reasoning of the 4 imams and the scholars who followed them. Because their individual methodologies in interpretation and extraction from the primary sources were different, they came to different judgements on many matters. For example, there are subtle differences in the methods of prayer in the 4 schools, yet the difference is not such that separate prayers need to be held. In fact, a follower of any school can pray behind a Imam of any school without any confusion.