all features of yEd Live are purely client-sided (except for the cloud services and GitHub uploads which must be triggered explicitly by the user).
Also note the About dialog of yEd Live:
You own your data: everything that you produce using this app, will stay yours. We don't even get to see your data. The data is stored in your systems (your hard drives, local memory, or your own cloud storage that we don't have access to) and your browser's local storage. Your files will never be sent to our servers or servers that we have access to unless you send them to us via email or actively via other feedback mechanisms.
Thus there shouldn't be any privacy concerns regarding yEd Live running in the browser. Because you don't upload your file anywhere, it always remains in your system.
Regarding the Graphviz import in yEd Live: This is pretty raw and rudimentary. It mainly supports the diagram structure and a few style properties, but all in all it supports only a very small subset of the Graphviz features.
There are currently no plans to support dot files in yEd Desktop.