This all depends on what social class your relatives were. Middle class and upper class Jews were pretty integrated into the German and Hungarian society, and they would have spoken German and Hungarian. It was the lower classes who spoke Yiddish.
Yes, I wish I knew more about their class status. I agree that is important here. All I know is that Johanna Hesseberg married a Prussian Jew named Adolph Fink, who spent his working life in Boston and Providence, R.I. as a clothing dealer, probably a tailor (finding “Adolph Fink” in Prussian 19th century birth records online is like going to the L.A. phone book to find your relative “Carlos Garcia.”)
The response to this is very interesting and a bit surprising to me. I would have guessed German as the “strong” language, with Hungarian spoken the way a border Texan speaks Spanish. It seems clear from the actual family histories that Hungarian might well have been the first language (depending on class), with Yiddish a secondary one (at some level: I’m interested in the discussion on this, too).
Eva Luna:
When I get more information on these people and their remaining family in Hungary and Prussia (a long term goal) I will go to databases like Yad Vashem’s to discover the family fate revealed there. I’m interested in this site you quoted as a starting point. Yes, I’ve had good experiences researching other branches of my family, discovering, for example, that I’m related to a large portion of the population of Nova Scotia and, separately, have living 2nd cousins in Manitoba, two things I would never have guessed. And then there were the family’s Quaker martyrs at the Restoration… I’m a pure bred American mongrel.
You may also want to check out the Federation of East European Family History Societies.
The large majority of Jews in Hungary spoke Hungarian or Standard High German at home rather than Yiddish (Middle High German).
From a religious perspective, the majority of Hungarian Jews belong to the Neolog movement. In terms of liturgy and mincha, it’s kind of Modern Orthodoxy. Men and women are separated but there is no mechitza and Hungarian is used in liturgy. From a cultural perspective, Neolog was always adamant about Jewish emancipation, which included the use of vernacular at home, which was Standard High German and Hungarian in the Austria-Hungary Monarchy. Although today Neolog may seem to be on the right side of the spectrum of current Jewish movements, historically it evolved from the same roots in Germany as the Reform movement.
A minority of Jews in Hungary are Orthodox or “Status Quo”. The use of Yiddish was more common within these movements. Definitely among Orthodox and sporadically among “Status Quo”. Please note that “Status Quo” includes a large variety of practices. In the US you would rather say non-affiliated communities.
Hungary might have been different, but my mother-in-law was from a wealthy Viennese family and did not speak Yiddish. She said they spoke an identifiably Jewish dialect of German. They left in 1939, penniless, and only because her younger sister was US born (the family had spent two years in NY).
My colleague who was born in Germany also speaks no Yiddish, even though his parents were from Poland. He left in 1939, age 16, and now claims not to feel comfortable speaking German. Although he certainly went to school in Germany, all his higher education was in English and he cannot speak technical German. He will be 91 in two weeks and is still writing papers.
For what’s it’s worth, my father in law was born in Hungary and speaks Hungarian (among other langauges).
Zev Steinhardt
Note that conversely, often people who grew up speaking a language in the home and another outside, or who learned a second language in school, are missing lots of “kitchen terms” in their outside/second language, lots of the words you’d learn as a non-professional cook, seamstress or decorator. ESL lessons include words such as “cheese”, “green peas” (), “bread” and may even include “loaf” or “fried”, but they won’t talk about “crust”, “shelling” or “basting”.
() which left us wondering what other kids of peas there were, and the teachers were never able to answer that question
Nava: there are yellow peas that are usually dried, in English dried peas are called “split peas”. “Green peas” usually mean fresh peas, or it could mean dried peas that happen to be green.
And I would be a perfect, erm, textbook example of that phenomenon in three different languages 
Hebrew was until recently, a dead language, reserved for reading the Torah. Yiddish, as mentioned, was the lingua Franca for European Jews for centuries, so, I’m gonna guess she spoke Hungarian (Magyar?) and/or Yiddish.
Interesting factoid: Szilard’s advisor was Einstein. They patented a novel form of refrigerator that had no moving parts.