Bismihi ta'ala
It is rather the opposite, as we can see that many Sunnis and Shia start the holy month of Ramadhan, and have Eids together on the same day. The issue is that in the Sunni world there is a difference of opinion as to whether the beginning of the lunar month can be pre-announced, or must it be eye-witness.
In Sunni communities there are those who follow Saudi Arabia's announcement, for example, and there are those who follow the local moon-sighting, by the naked eye.
For further statistics on this, please review the following site:
https://www.moonsighting.com/1440shw.html
Wassalam