I know this sounds mainly like a linguistic question but I'd imagine that official document surviving from that time and chroniclers would give us direct or indirect answer.
Did a firsthand source, say, mention offhand that Charlemagne spoke to a Saxons count directly in one instance, and mention he needed a translator to talk to a Moorish embassy in another, with the implication being that Charlemagne didn't need a translator to talk to a Saxon?
Did everybody just speak Latin to each other?
I know this sounds mainly like a linguistic question but I'd imagine that official document surviving from that time and chroniclers would give us direct or indirect answer.
Did a firsthand source, say, mention offhand that Charlemagne spoke to a Saxons count directly in one instance, and mention he needed a translator to talk to a Moorish embassy in another, with the implication being that Charlemagne didn't need a translator to talk to a Saxon?
No records survive of Charlemagnes personal or private correspondence, and it is not at all certain that this kind of correspondence existed to begin with as Charlemagne was not very literate. His biographer, Einhard, mentions that Charlemagnes handwriting was very bad and that he learned to read while already an adult.
In any case, diplomacy in Late Antiquity and the Early Middle Ages were not a written affair. Written instructions or communication was not common during the 8th and 9th century in Western Europe. Rulers send out legates or envoys with set instructions to foreign states, tribes or monarchs and it was not uncommon for the king to conduct his own diplomacy in the field. In the case of subjected peoples, a count or duke would be installed from the monarchs personal entourage as a proxy between the people and the king; which would then be periodically checked upon by a kind of traveling court system. Charlemagne ruled his empire, by traveling from place to place in what is called an itinerant kingship: he basically needed to be there, in person, with his entourage to rule effectively.
Did he use Latin? Were their mother languages still intelligible?
His biographer Einhard mentions that he also spoke Latin and understood some Greek, but one shouldn't expect too much of this claim. Apart from that, the amount of people among the Saxons proficient in Latin would have been extremely small.
As for the intelligibility of Charlemagnes Germanic dialect and that of the Saxons; it's often assumed that he could communicate with his Germanic-speaking subjects, but current linguistic consensus is that this would have been quite difficult.
The Saxons spoke a North Sea Germanic dialect, whereas Charlemagne most likely spoke a late form of Elbe Germanic with advanced signs of the Second Germanic consonant shift; ie. most closely related to modern German. These variants, though still connected geographically, split from a relatively uniform South Germanic form some 400 years earlier. Communication would have been possible, but it would have required an interpreter or someone familiar with both variants to get into any complex discussion. The Germanic language family is quite divergent compared to, for example, the Slavic or Romance branch of the Indo-European languages. It's always tricky to compare modern languages with ancient ones, but it is not unreasonable to assume an overall intelligibility akin to that of French and Italian or Polish and Ukrainian. This is especially true for Old Saxon and Old High German, because modern Low German (which, in its core, descends from Old Saxon) and modern German have converged to a very large extent over the past 10 centuries; and by converged I actually mean that Saxon/Low German became increasingly more like German.
(For those interested; the previous Merovingian dynasty on the other hand most likely spoke a late variant of Rhine-Weser Germanic, which is most closely related to modern Dutch rather than modern German)
Did everybody just speak Latin to each other?
No, knowledge of Latin tended to be limited to the clergy; and only the higher clerical members at that. In any case, it was primarily a written language; not a spoken language.
Nobody really spoke Latin prior to the Renaissance, when the Catholic Church and Humanists reconstructed its pronunciation largely based on contemporary Italian. It was read out loud in sermons, but it wasn't really used to converse with one another.