Pidgins don't actually arise that way, instead, they depend on repeated but intermittent short-term contact between two groups. For instance, Chinook Jargon arose from casual trade contact between various tribes of the Northwest, where there wasn't enough long-term contact for one side to learn the other's language.

It is expected that there could be substantial influence between Hungarian and Slovak, so that a Hungarianized dialect of Slovak could develop. Countering this, though, are normative pressures to conform to the standard language. In fact, the situation there is so complex that I question the possibility of getting an accurate answer. There are legal and social pressures for Hungarians in Slovakia to Slovakify, and countervailing social pressures to Hungarify. (And vice versa in Hungary, though to a lesser extent as I understand, since there are many fewer Slovaks in Hungary and they are more dispersed). The only reliable research method would be to surreptitiously record conversations between known individuals, and observe the behavior of pure Hungarian-on-Hungarian interactions, compared to Hungarian-on-Slovak interactions. (Fluent bilingual people switch languages quickly when a monolingual or marginally bilingual person joins – may to include the person, maybe to exclude them).

The hypothesis that one would want to test is that Hungarians in Slovakia all speak "regular Hungarian", but may increase Slovak features when a Slovak speaker is added to the conversation. There are very many outcomes possible from this grand experiment. The problem is that there is a lot of personal knowledge stuff that has to be controlled for, such as whether Sally hates Hungarians or not, whether Sally is hated by the Hungarians, how fluent Sally is in Hungarian, etc.

If one were a linguist growing up in the Hungaro-Slovak region, one might specific properties of the local Hungarian that can be attributed to Slovak influence (or the other way), and then perhaps devise a very subtle method for measuring that feature. The other problem is, though, that there is no such thing as "regular Hungarian", so one would need to compare a suspected feature with what is found in relevant dialects in Hungary. Northern dialects of Hungarian seem to be quite understudied.

It would be really interesting to know all of this, I just don't see any way that the experiment could actually be conducted.

I have little to add to panini's excellent discussion above, except to also point out that bilingualism can be stable and maintained in certain situations. This sounds like one of them. There actually are bilingual or even trilingual(+) villages in the world where it's simply the norm that people shift between languages for daily communication. In the right circumstances, the result may be stability rather than eventually shifting to just one language for the whole village, or creating some sort of mixed variety. I know I've seen examples like this from reading about different communities in passing, but I can't list them at the moment (I think there are some cases like this in India and Australia?). If you're interested in the topic, then that might be one way to indirectly research the topic.