From what I could gather the other day Sturgeon could hold her own referendum within Scotland, but as it would not be an officially UK sponsored referendum it would carry no weight in Intrernational law. I might be wrong, but that was how it came across.
What she could probably do with it, if it was transparent and both sides got equal air time and finance etc. is take the result down to Westminster and try to make a case for it to become law or for another UK sponsored Scottish Independence referendum to take place. Whether the UK would be open to any of that is to be debated.
Where she has other problems is with timing and EU accession. If Brexit is not complete yet Scotland gains independence it would not automatically stay in the EU. That was made plain over the weekend by the EU. If it applied to join, and somehow jumped the queue, the UK, still a member, would most likely veto entry. So might Spain.
If Brexit is completed and Scotland gets independence, and gets entry to the EU, it would have no choice but to adopt the Euro and join Schengen, among many other things. The UK would almost certainly then have to impose border controls. I know that when Brexit is finalised the UK will have a land border with the EU via NI - Ireland, and it is expected that a treaty to keep that border 'borderless' will come into force. The reason for that being the case is that Ireland, as with the UK, is not in Schengen. Scotland in Schengen changes the game.
The whole thing is a bloody mess, but that is what was voted for and we have to get on with things as best we can.
Sorry Suff, I was composing as you posted, but our ideas seem roughly the same.