Traditionally, the formal teaching requirement at universities in continental Europe is the so called Habilitation. It's a formal exam which requires submitting a thesis and defending it in a colloquium much like the PhD, but (supposedly) on a higher level. Usually it also includes a teaching-related element. The habilitation used to be a requirement for being appointed as a professor, but in recent times also other criteria, which represent a comparable achievement to a habilitation, are being used. In Germany, having passed the habilitation allows you to use the title "Privatdozent" (even before being appointed as professor), with the same teaching entitlements as an ordinary professor.
However, also other personnel may teach at the university. Even fresh graduates from one study program may find themselves in front of a class of students who could be just a year or two behind them. However, that will always be under the supervision of a professor, or on the formal commission of the department dean.
In addition, there are recent efforts to offer a more formal teaching education for university staff, including certificates for those who take part in the programs. I'm mostly familiar with the situation in Germany, for example this or this (in German), but I imagine similar efforts exist in neighboring countries. At the moment, these certificates are however not a requirement for university teaching, and I even doubt that they will become so in the near future. Yet, in some announcements for professor positions it seems to be included as a desirable applicant qualification.
