Well usually your PI will do the initial stages of networking or carry enough weight that you don't have to do much networking on behalf of your publications.

Well usually your PI will do the initial stages of networking or carry enough weight that you don't have to do much networking on behalf of your publications. That is a separate question from if your PI will do networking FOR you. Some will, some won't, either by malice, by ignorance, or by temperament. I worked for a nobel laureate (who discovered restriction enzymes). Publication was not a problem. He was lovely and fun to work for, but in the end I had to leave academia because he couldn't be bothered to network for me (he was more interested in playing penny slots at the casino). If you're in grad school, my strongest recommendation is to optimize first for a PI that will network for you. That is generally the most valuable form of networking, because 3rd party validation generally carries a strong amount of weight by human nature. Plus it potentially takes minimal effort on your part.