So my, albeit limited, understanding of the Huns is that whether or not they were a concrete tribal or linguistic identity at one point is uncertain but that by the time they reached Europe, and later the territory of the Western Roman Empire, they were more an umbrella term applied to many groups that may have had little in common other than nomadic lifestyle and raiding Roman territory?
There were specific tribes that existed and spoke the same language such as the Vandals, Goths, Sarmatians, Alans, Franks/Frankii etc., but to my understanding, the term Hun can be broken down into two distinctions:
Huns - A tribe which originated from the Asian steppes around modern day Kazakhstan, Uzbekistan, Turkmenistan, Western Mongolia, and Russia. Otto Maenchen-Helfen claimed they had their own language that resembled Proto-Turkic and were a unique tribe from the umbrella term 'Hun' that we would recognize in the historical sense.
"Huns" - This is that larger umbrella term referenced above. They involved many different tribes made up of Ostrogoths, Huns, Alans, Sarmatians, Germans, Franks, etc. and they each spoke their own languages. Priscus states that the "Huns" spoke a combination of languages. These languages ranged from Gothic, Hunnic, Proto-Iranian languages, and Germanic languages. The ancient source claims that Attila spoke a different language from the Turkic Hunnic language in his court. The various tribes under Attila combined Gothic, Hunnic, Latin and various Germanic languages based on their exposure to Western Roman traders, but hardly any would've spoken Greek due to limited exposure to Greek speaking Byzantines.
In this theory, yes, any given warrior riding with Attila would've been able to understand their orders and would've most likely been able to communicate, but the 'Huns' of Attila were comprised of multiple tribes and various ethnic identities. I would imagine a sort of uniting colloquial language comprised of a combination of borrowed words from Hunnic, Latin, German, Gothic, and various other plains languages.
Ah it appears that I missed this. Although /u/karnesman has touched a bit on it, the answer is a bit different from his view.
The Romans used the term Hun in two specific ways, as shown in Procopius' description of the Hepthaltites, who he called "of the stock and the name of the Huns..." There was an actual group of people who identified themselves as Huns - that being the old Xiongnu, but also a broader "stock" of Huns, which fell under the more generic usage of the term. This primarily referred to people who spoke a branch of Turkic called "Oghur" or "Lir" Turkic and included Huns, the early Bulgars, Avars, the Kidarites/Chionites (who were Huns proper), and the Hepthaltites. Later this was expanded to include Gok-Turks, Khazars, and other peoples as the term "Hun" became as generic as "Scythian" in the 7th/8th century. Alans, to one extent or another, were probably always inadvertently included under the term.
The other problem resides in the use of the Hun identification by non-Hunnic groups. The Huns themselves were affiliated with the proper imperial identity of their Xiongnu predecessors, meaning any group who wanted power would also want to take up this identity regardless of whether they had any actual relation to the old Xiongnu dynasty. We have a clear example of this, where a group of Oghur speakers who were present on the Volga shortly before the Hun invasion, and retreated in the face of the Hun advance (the Alpidzur and Tongur) took up this title. The Kutrigur and Utigur Hunno-Bulgars, who were established in the Pontic from 480-558, also did, although there's a pretty solid argument that they were directly connected to Attila's dynasty and probably a continuation of the Hunnic state (that is still conjecture).
Sort of the same concept as the Holy Roman Empire taking on the moniker of "Roman Empire" and Charlemagne taking up the title of "Roman Emperor" despite the fact they had absolutely nothing to do with the Roman Empire (which at the time still existed, the so-called "Byzantine Empire" by modern historians). Or how we call the Tartars "Mongols" despite the fact they were a separate tribal grouping. Or all the various "Turkic peoples" of the medieval steppes (in fact the Chinese call the Ashina/Turks "Xiongnu/Huns"). We call the modern Turks "Turkic," because of their language, but their actual ethnicity is closer to Anatolian Greek or Armenian.
Identity is a very difficult topic, especially in regard to the Eurasian steppes.
As for Language, Priscus states that Gothic was a "lingua Franca" of the Hunnic Empire at the time of Attila. We know Latin was also fairly well known, Greek much less so. Hunnic was also widespread, but evidently Gothic was the most common. In reality there were multiple Gothic dialects, so these peoples probably could relatively understand each other but still faced difficulties in communication between different groups at times. Gepid (a people who spoke Gothic) dialects would have expressed differences from the Amali ("Ostrogoth") dialect, for example.