The pertinent rules are 9.7 and 17.5.1
My new book in enroute so, if there have been any changes, this may be dated:
If you take all 10 (or 15 or 20) during regular match firing in HP or BP, you may opt to continue immediately or wait until the end of the match. (Everybody else waits until match conclusion.) If more than 30 minutes has gone by since your original string completion, you are allowed a sighter. That used to be mandatory but is now, I believe, optional. If you choose to shoot a sighter, it doesn't count toward your score -- although I'm told that it was at one time 'convertible'.
I see nothing in the book about any rest period. IMO, if you take more than the normal 'stand easy' time between strings, you'd be out of line; but I'm not certain that that's correct.
It can be done at any NRA sanctioned -- approved or registered -- match.
Yes it has to be witnessed! By at least an appropriate match official. (Let's get serious once in a while! Do you really think that anyone will accept a 'National Record' that nobody but the shooter noticed happening?)