I develop on multiple machines and have just been through this exact thing with both VS Code and Pycharm. I was surprised at how different the implementations are. The VS Code SSH extensions work great - I would almost say "seamlessly", actually - whereas the Pycharm approach feels like a real kludge.

For starters, there seems to be more than one way to do it - one of the responses below refers to remote-mounting the filesystem using SSHFS, but another way to do it is through the Tools -> Deployment menu. The SSHFS approach is presumably the better one if your particular problem is trying to write code across several machines; the Tools -> Deployment avenue seems much better suited an actual deployment (it is, after all, in the name).

It's got me wondering whether I could get away with doing all my Python development in VS Code but obviously everyone has different constraints.