Did they refer to one another as Saxons or were there other terms? Did they consider one another related and have any significant or noteworthy contact or were they indifferent to one another?
There were certainly contacts by the late-seventh century since 'Anglo-Saxon' (I don't object to the term but here I'm using it for convenience not accuracy) were aware continental Saxons were not Christian and sought to send missions to convert them. There seems to have been an active area of trade in the seventh and eighth-century North Sea that linked these groups together.
The 'Anglo-Saxons', or at least Bede who would have called himself English (he self-describes as an Angle, which is simply the nominative form of the noun (now obsolete) from which we derive the adjective English), called the continental Saxons either Saxons or Old Saxons (distinguishing from those who came to Britain in his account). The lives of various Anglo-Saxon missionaries call them Saxons, but most of these were written on the Continent and all by people with good links to the Frankish kingdoms.
We don't know what the continental Saxons called the Saxons in Britain, since by the time literate Saxons paid attention to Britain (the late-tenth century) a kingdom of England ruled all the Anglo-Saxon territories other than Lothian (in the kingdom of the Scots), so they talk about the English, but we can't say this was the terminology used earlier. The Franks used Saxon indiscriminately of continental and British groups, to the point where the Frankish queen and saint Balthild who was apparently born a Saxon (and was a slave early in her adulthood - she had an interesting life) can equally well be argued to have been from Britain or the north-west of mainland Europe, and therefore frequently is. In the mid-late ninth century the Annals of Saint Bertin (at this point probably written in the court circle of Charles the Bald) do start a trend of using Anglo-Saxon to describe the West Saxon kingdom that was hegemonic in southern Britain, probably because the Saxons on the continent were part of the kingdom of Charles's brother and often rival Louis the German. We don't know if this was the practice in continental Saxon areas though.
You might be interested in some earlier threads about these peoples: