Pidgins generally arise between interactions with merchants, traders, and the like, while the elites would be far away in the capital, not haggling for fish with the foreigners. They tend to simplify their phonology, morphology, and vocabulary significantly.
In my world, the situation causing two groups to interact is quite different: there are humans and demons (think standard D&D Tieflings rather than metaphysical spirits), who live in different dimensions. Humans have figured out how to summon demons, but unfortunately they can't actually magically compel a demon to do their bidding. However, both races recognize that a cultural exchange would be valuable, so they are willing to engage in bargains. Here are some details about the situation:
Typical magical high fantasy setting, both races have roughly equal technological/magical advancement, but they specialize in different schools of magic.
Only mages can perform a summoning and be summoned, and mages are an elite class in both races, so the participants are always high-class and well educated.
They are interested in the other race's knowledge and secrets, both arcane and scientific. So they need to be able to convey complex knowledge, and conduct very elaborate bargains.
(The exchanges are not always purely about knowledge, they might also give resources or perform magical feats in exchange for knowledge.)
Both sides are also motivated to reveal as little as possible to the other side. They want to conserve fodder for exchange, and they don't want to give the other race too many advantages in case of an invasion. They would not let the other race send over an envoy to learn their language.
I envision that the interacting mages will devise some kind of pidgin/hybrid language. My question is, what kind of features might this language have, and where would it be similar or different to real world pidgin languages?