If a financial institution receives a FinCEN 314(a) search request on the 10th of the month and doesn't complete the search until the 24th of the month (the same day the next FinCEN 314(a) search request is received), is that a violation or is that still within the allowable 14 calendar days?

14 days is 14 days regardless of the next request date. There can be emergency transmissions at any time and they do not always come every two weeks.

I'm hearing differing opinions as to whether that would be considered 14 days or 15 days. How do the regulators view it?

The financial institution must search its records and report any positive matches to FinCEN within 14 days. You get ther request on the 14th, you must report by the 28th.