In our community I am the only one who follows actual sighting of the moon for determining the Islamic months.

Answer

As far as Eid is concerned, one or two persons cannot celebrate it alone in isolation of the community. The Shariah has guidelines and rules for all situations.

(1) If you alone in the whole community see the hilaal of Ramadhaan, but your testimony is not accepted by the Jamaat of your place, hence officially for the community the next day will not be Ramadhaan. Nevertheless, you should fast the next day.

(2) In the scenario mentioned in No.1 above, there is the probability that you will have to fast 31 days because it is possible that at the end of the community's 29th day (which is your 30th day of fasting) the moon is not sighted. Hence, the next day will be the community's 30th day while it will be yours 31st Day. You are required by the Shariah to fast this 31st day. You cannot celebrate Eid alone.

(3) Assuming that you alone saw the hilaal of Shawwaal, but your testimony was rejected and the community regards the next day as the last day of Ramadhaan. Then too you have to fast and believe that there was an error in your sighting.

(4) As far as Muharram is concerned, fasting on the Day of Aashura is not a communal act of Ibaadat. If you saw the hilaal of Muharram, but not the others, then you may act according to your sighting, and fast when it is the 10th according to your calculation. (5) The same explanation as No. 4 applies for Sha'baan.