Who should create contacts with other universities/departments/researchers during a PhD study? The supervisor or the student?
- The supervisor will already know plenty of people and help the student establish contact when it is on topic.
- The student should just as well try to get in touch with other researchers who might provide opportunities for collaboration, depending on the context the student is in. For instance, the student has particularly good chances for that when being funded from a project that involves researchers from different institutions, or when attending conferences in fields where conferences are commonly held and visited.
How these contacts may affect the continuing of the academic carrer of the student?
They provide opportunities of getting in touch with different fields, possibly exchanging some data that can be used in the other's research, and learning how to autonomously organize little "projects" (by this, I mean e.g. studies and ensuing papers).
May be possible that a specific contact during the PhD will evolve in a post-doc position later on?
This is one of many theoretical possibilities, though probably not one you would normally strive for in most cases.
Should contacts always be made?
This is somewhat vague, but as a general rule of thumb: If there is an opportunity to get in touch with another researcher, at least have a brief conversation to check whether the two of you are "compatible" to start with.