Does Markdown Journal ever store your files on the server?
No. The files are not held on the server.
Your files are stored in your Dropbox account. They are temporarily downloaded into a memory buffer for the purpose of appending new content and re-uploading back to your Dropbox folder, but the contents are never stored on the disk.
Markdown Journal does not log or store your entries. The application code also does not track your usage in any way, shape or form. Please note however that your usage is logged by the server for maintenance and security purposes. These logs are never shared with any third party and are only used internally for the purpose of security audits and usage metrics.
Why should you trust Markdown Journal with your personal diary? You shouldn't. As a rule of thumb you should never fully trust any online service with your data. You shouldn't even trust Dropbox.
But in case of Markdown Journal you can actually verify it is not doing anything shady by reading the code.