An archivist at the National Archives of Hungary has been assisting me and there has been a fee for this that covered the researcher's time, translation of records and postage. Mostly he has found records which aren't available through LDS, but there have been a few I could probably have found myself. I haven't been researching long and am only now getting the hang of accessing the microfilmed records! However, my understanding from looking at their website is that you can go into the archives an do your own research - and it's free. The link to their website is http://www.mol.gov.hu
. I don't think that you can search for family records online. Also, there's currently a message on their website that the research room in the main building will be closed from August until December so it may be worth checking this out.
I hope this helps a little. Maybe you'll find out something about my CSONKA whilst you're researching your own! Were they from around the same area?