I am an estate executor. I'll file a 1041. Should I send my attorney an 1099-misc for statutory fee he received ?

I cannot think of a case where I have done so, but instead documented the expense against the estate using the attorney's letterhead invoice.

Don't forget that you will have to report on line 21 of schedule 1 (formerly on the 1040 itself) your receipt of expenses as Personal Representative (previously Executor) to the estate.