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.