Can anybody with my basic information, find out if I filed my tax returns. If they have my social security number and name and maybe date of birth, can anybody know ifi filed my tax return?

With name and SSN, date of birth and perhaps the address on the return, YES, they could call the IRS, convince them they are YOU, and simply ask about the status of your return.

Also, if 2 people claim the same person (such as a child), the IRS will contact both parties about each other.
Do you have a specific worry about all this?