Yes, the second screening occurs after you get the LoAs. Basically, the MEXT home office in Tokyo conducts a similar review of applications to what the Embassy already did in the preliminary (first) screening. The purpose is to detect any unexpected problems or discrepancies in your documents, that might have been overlooked by the Embassy. Next, the MEXT home office will contact the universities on your list (refer to previous posts about MEXT skipping private unis in favor of national unis). After finding the right university, MEXT gives everything a final look and then makes the decision (final screening) -- they announce the results through the Embassy.
So for me, I passed the second screening by MEXT and they told me in late-November. Then, they only contacted Tokyo Tech (my 2nd choice) and decided to place me there. Finally, the Embassy informed me mid-January that I was placed at Tokyo Tech, and thus had passed the final screening.
Good memory! Yes, I fly to Tokyo on Sept 27...very excited!!

If the April start-date had worked with my previous graduation schedule, I would have taken it...but I had to wait until October. It works out better this way for me, with all that I need to prepare...
Good luck to all waiting for the second screening!