There are two main types of vocational training in Germany (Ausbildung). There are ones where you train solely at a school (e.g. elderly care, nursing). These you can do while in the Dublin procedure.
The other type is dual vocational training (duale Ausbildung), where you learn partly in a school and partly by working directly at a company. This type is considered as work and in order to work asylum seekers require the permission of the immigration authorities. For the first 9 months (in Dublin and non-Dublin cases), asylum seekers are not permitted to work.
After the 9 months have passed, the situation is as such:
According to the German asylum law, it is not possible to work (including a dual Ausbildung) when another EU country has been deemed responsible for the applicant as part of the Dublin procedure. This is according to paragraph 61, section 1, number 4 of the asylum law (here via online translation):
However, there was a decision at the European Court of Justice that says a general ban on working for people in the Dublin procedure is not permissible. As a result, asylum seekers in the Dublin procedure must also be given permission to work after 9 months. Here is a link to a short report on the court decision (via online translation):