Danelaw in East Anglia, Northumbria and eastern Mercia did not result in a homogenous society, neither was it truly bi-lingual. The result was two speech communities in Danelaw living beside each other, one speaking Old Norse and the other, Old English. The real question is: could these co-inhabitants of the Danelaw region understand each other (i.e. mutually intelligible).
I am not sure if it's now a settled matter by historical linguists and historians.
As I am not up-to-date on the latest research, I will provide an older reference. From Language and History in Viking Age England - Linguistic Relations between Speakers of Old Norse and Old English (Brepols, 2002). pp 2-3:
... it is also clear that the native Anglo-Saxon population in the
areas of Scandinavian settlement was by no means driven out or
otherwise suppressed; and so in the tenth and eleventh centuries
Anglo-Saxon England is more properly to be regarded as
Anglo-Scandinavian England, with the two peoples, similar but
distinctive, in close and persistent contact. Sir Frank Stenton, in
his classic review of Anglo-Scandinavian relations and the settlement
of what he termed 'the essential Danelaw' (that is, between the Tees
and the Weiland), concluded as follows (1927, 241, 246: 'The Danes in England', Raleigh Lecture on History, Proceedings of the British Academy, 13, 203-46):
[W]e begin to discern two races in pre-Conquest England, differing in
language, law, and social order, held together by little more than
common acquiescence in the role ofa king whose authority was narrowly
limited by custom. We are driven, in fact, towards the conclusion that
the superficial unity of the Old English state concealed a racial
cleavage which was none the less real because it was taken for granted
by contemporaries. [...] All lines of investigation - linguistic,
legal, and economic - point to the reality of the difference between
Danes and English in the tenth century.
Further down, still at page 3:
The crucial concept here is that of the 'speech community': speech
communities may or may not correlate with other types of social
community, but at the very least one can easily distinguish the Old
Norse and Old English speech communities in Viking Age England, and it
is for this reason that linguists have been more hesitant than, say,
archaeologists and historians to embrace those contemporary ideas
about 'ethnicity' which substantially downplay the importance of
linguistic factors in the creation and maintenance of group
identities. There is no need for a full rehearsal of the issues at
this preliminary stage; but as will become clear, one of the outcomes
of this book will be to uphold broadly Stentonian perspectives on the
reality of the English/Danish distinction in Viking Age England.
More recent books on this topic:
- Viking Age England (History Press, 2004)
- Everyday Life in Viking-Age Towns - Social Approaches to Towns in England and Ireland, c. 800-1100 (Oxbow, 2013)
- Viking Age Yorkshire (Blackthorn Press, 2014)
It does appear the argument and evidence of mutual intelligibility persists. So, these 'speech communities', although not truly bi- or multilingual, could communicate well enough to get by because of the common root of Old English and Old Norse. From Peter Trudgill's recent book, East Anglian English (2021), at page 13:
But from about 890 onwards, East Anglia became officially part of the Danelaw – the area of England which had been signed over to the Danes by Alfred, the King of Wessex. After the East Anglian defeat, parts of East Anglia were “shared out to Danish Viking soldiers, who were thereby transformed into settlers” (Nielsen, 1998: 167); large numbers of further settlers subsequently arrived from Denmark in a secondary wave.
Prior to 869, the Anglian-origin English language of East Anglia had developed gradually, with little outside influence, for about four hundred years, but now the area became bilingual again, although there remains the interesting question as to what extent bilingualism as such would actually have been necessary. The original parent language of Old Danish, North Germanic, was a close relative of West Germanic; and indeed many linguists postulate an earlier language which was ancestral to both, Northwest Germanic, which would have split up into North and West Germanic around 450 AD. ... It is therefore quite possible that English speakers and Danish speakers could still understand one another reasonably well during the 9th, 10th and 11th centuries even without becoming particularly bilingual as individuals.
From Wikipedia, "Peter Trudgill, FBA is an English sociolinguist, academic and author. He is a member of the Norwegian Academy of Science and Letters, and a Fellow of the British Academy."
Nielsen, Hans Frede. 1998. The continental backgrounds of English and its insular development until 1154. Odense: Odense University Press.