Shalom!
  In response to the question regarding the timing of the celebration of the New Moon (Rosh Chodesh) and other feasts, yes the original way was by direct observation.
 
 Unlike the astronomical 'new moon' which is invisible, the Hebraic 'new moon'
or new month began when the first sliver was sighted and correctly verified to the Sanhedrin by two reliable witnesses.  Signal fires were then lit to alert the people of the official beginning of the new month, and a holy convocation unto Adonai.

  Fast forward to the diaspora and the modern era and we see this method is not practical.  As Jean points out, there is no one to 'report' to, and we are all not living in Jerusalem or even the Promised Land (Israel!).

  The development of the calendar by Hillel averted this problem.  An excellent
explanation is given here:

http://www.jewfaq.org/calendar.htm

   But, what of the essential question?  Suppose we are 'off' (which is inevitable)?

  The best answer was given to me by another Messianic Jewish brother.  He pointed out to me that Yeshua Himself kept the festivals at the times the Children of Israel celebrated them.  (One might argue that Yeshua celebrated the Pesach a day early, but that was necessary to stay in alignment with HE BEING slain exactly at the same time the Passover lambs were also being slain.

  This year, our calendars show that Erev Rosh Chodesh Tishrei; Yom Terurah; Rosh Hashanah begin at sundown 12 September.  (Remember this is 13 September to the Jew, for a new day begins at sunset the previous night; hence sundown September 12th ends that day, and begins the 13th, for secular calendar comparisons).

  All this writing to arrive at this simple answer!   We do our best as a people to
meet HaShem at His appointed times.  Knowing that we do this as closely to His
commandments as we can, will He not be pleased with us?

  Would a husband reject a wife who was a few minutes 'off' from meeting  him for an appointment?   In the case of the Feast of Tabernacles, which lasts 7 days plus 1 more day, consider this:  Suppose you were to meet your beloved
at such and such time to depart for a week's vacation.  It is just the two of you, going off on maybe a second honeymoon!  If you were a few minutes early or late in meeting him, would you leave or would he leave without you?

   He loves you!  Yeshua Loves Us!  We are His Bride, and as His Bride, we do our best to check and double check our schedule, for we are also so in Love with Him; we cannot wait to be near to Him again.

   Are we prepared everyday?  Yes, the appointed times are very special and Holy convocations unto the L-rd, yet if we are ready every moment, then are we not like the Five Wise Virgins?

   Let us be not concerned then, just prepared, ready to meet our Spouse, Yeshua Hamashiach, for soon the call shall go out

"Behold, The Bridegroom Comes!"

In His Unfailing Love,
His servant and Bride,
~yochanon

"Wait on the L-RD;
         Be of good courage,
         And He shall strengthen your heart;
         Wait, I say, on the L-RD!
-Psalm 27:14