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The designer, within technical constraints, e.g. available materials. If you make prototypes in pool production, the manufacturer often uses a standard stackup which is described on manufacturers web site.
As a pcb designer, you are responsible for the design of the board. The PCB stackup is a very important part of the design. As a designer you know all the requirements for your board. And yes, there are a lot of designs where the stackup is not that important and the default stackup of a pcb manufacturer will do. But that is your decision !!!!!!
For 99% of the boards I design, a customized stackup is needed. In such cases;
ALWAYS communicate with your board manufacturer.
You know what you need. They know what materials they have and what they can produce.
Stack-up design is a part of PCB design, Trace widths/spacing for your impedance controlled lines, the current-carrying capacity of your planes, and many other things depend on the thickness of the PCB material and copper thickness used. At the same time, this stack-up should meet the DFM checks of the PCB vendor.
So better approach is to discuss with the PCB fabricator in the very initial stages of the design and reach in a stack-up which is OK for both designer and fabricator.
I never submit a PO for boards before having a stackup approved by the vendor, including a document which includes substrate material definitions, copper thicknesses, and imedance requirements (if any).
The next time I need that stackup with that vendor, I'll just cite the original stackup specification. If I change vendors, I start the process over.
The only exception is when I'm doing something very basic where impedance and current handling aren't very important. In such cases I'll let the vendor decide which of their "standard" stackups to use.