Hi Mike, thanks for reaching out. Lucidchart is a cloud based software so everything from working on document to storage is done via cloud services. You can download your documents as PDF, PNG, JPEG, or VDX files however you won't be able to edit them unless you're connected to the internet.
We have an Chrome offline app that can be used to work on documents offline: https://www.lucidchart.com/pages/offline
Here is a good community post with some Visio import tips!