The former would be preferred for more serious works of literature. Starting a sentence with or is very informal and not something you should do in an essay or a professional context.
The latter is a good example of something you would say, but not write. That being said, it is also a rule that is commonly broken and therefore accepted to be correct as well.
So, both are good! Just beware of context.
Would you mind providing some details, please?
Is perhaps the more 'correct' and slightly more formal way of rephrasing your question. Or the example given by @unorthodox grammar is just as good, and slightly less formal:
If you don't mind, would you please provide some details...?
The please is optional in either place, as the 'would you mind?' conveys enough politeness in my opinion, but it wouldn't hurt to add it in. It would usually come immediately before the action that you are asking the other person to do, i.e. please provide... or tagged on to the very end of the question
If you have quite a long sentence, the 'please' can be placed at the end of the clause where the actual request is made – to avoid waiting for the end of a very long sentence:
Would you mind providing some details please, for us to capture the steps required for future requests?
Alternatively:
Would you mind? Could you (please) provide some details...?
Is a bit more conversational. (Not something you would write in a formal email). Both of the examples you gave I have seen written or heard in everyday conversation, and yes, they do look/sound slightly odd constructions.
If you are making the request in an imperative, but still polite manner, the 'please' can come at the beginning of the request:
Please provide some details for us to...
Best Answer
Or you could simply, and more to the point, just say