When should I use
For example, this sentence:
We do not gather any private nor personal details.
nor here correct? is using
or also correct?
Nor is no longer used as frequently as it once was, (Ngram nor), but 'nor' is usually tied to 'neither' in sentences such as '...neither A nor B', but even in this case it is not uncommon to hear 'or' used instead.. Neither is also less frequently used than in the past, (Ngram neither), which has possible contributed to the lower use of 'nor'. One of the reasons for the decline in these two words is that we no longer speak as formally as we did in the past, e.g.:
In the past we might have said:
We saw neither Gwen nor Eric during our trip.
we are now more likely to say:
We didn't see Gwen or Eric during our trip.
The following link shows when grammarians expect 'nor to be used (3 ways to use nor). These rules should be applied in formal or academic use, but it is not uncommon to see/hear most of these rules ignored in less formal discourse.
In everyday English today, your sentence would be rendered as:
We do not gather (any) private or personal information.
Although, as I have indicated, I think that the word 'any' is unnecessary. I also concur with @FumbleFingers that I am not sure why you need to specify "private" as well as 'personal'. I would think that 'confidential' would cover both categories. This would leave you with the following, much shorter statement.
We do not gather confidential information.