ShareX is a beloved, free, open-source Windows tool, and it deserves the praise: it captures anything, automates workflows, and can upload to dozens of destinations - including your own FTP server or cloud. If you want maximum control over a capture-and-upload pipeline, ShareX is hard to beat.
So this is not an "own storage vs their servers" story - ShareX can absolutely use your storage. The honest difference is scope and simplicity: ShareX is a dedicated, deeply configurable capture toolkit; FTPie is a full file manager that happens to include capture and one-click sharing, so the screenshots and recordings you take live in the same app where you manage all your files.