On the relation between gross output and value added production functions (and why it matters)

The estimation of production functions remains ubiquitous in economics, both at the firm and aggregate level. While some focus on gross output production functions (GO-PF), others use value added production functions (VA-PF). Because value added is constructed as gross output minus intermediary inputs (properly deflated), VA-PFs are but a transformation of the GO-PF. Therefore, regardless of which is the starting point of researchers, one function is implicitly imposing restrictions on the other. Which are these restrictions? Are they reasonable? Does it matter (e.g. for TFP measurement)? I address these questions.