In the event of a vote for independence, decisions about UK citizenship would rest with the UK Government; the basis for entitlement to Scottish citizenship would be for the Government of an independent Scottish state to decide.
The UK has historically been tolerant of plural nationalities, and therefore it is likely that it would be possible for an individual to hold both British and Scottish citizenship. However, under current rules, British citizens living outside the UK cannot pass their British nationality on more than one generation. So, the children of British citizens living in an independent Scotland would be British citizens, but their children and subsequent generations would not be.
The Government of the continuing UK would also need to consider whether all British citizens living in Scotland could retain their British citizenship upon independence. This cannot be guaranteed and could be dependent on any residence requirements or proof of affinity to the continuing UK. It is not possible to predict now what the decision of a future Government of the continuing UK might be in this area.