Well, I wouldn't use such a website anyway (especially a document converter -- that is better done using a real application), regardless of where the processing was done, unless I was very certain that the website was trustworthy. For one thing, even if the website purports to not move my data to their servers, how do I know they're being truthful without going to extremes such as sniffing traffic?
There have been plenty of sites that have lied about such things.
> What I have for native applications that I don't for the web is the ability to firewall off the native applications.
There you're placing trust on the firewall's sandbox. Are you sure the application can't communicate with the outside at all? DNS exfliltration for example?
A firewall is not a sandbox, but yes, I am sure that the applications can't communicate with the outside at all. My logs would show if they were. Any and all packets that originate from them are dropped, including DNS lookups and the like.
You trust uploading your personal information to a server to generate the pdf/image/whatever vs doing it in solely in the browser?
Doing more on the server would lead to more tracking, not less.