In Automation testing, do you think it is important to have someone specialized in writing test cases and another one in writing automation script?

unless a conflict of interest could arise then no would be my answer.
Would it not make better sense to have the same person doing both?

It is not necessary both fields are different but at least we should update as our industry is growing day by day.

IN MY OPINION :- both of them are joined together even if each one just understands the consequences those the are being found according to the other

Automation testing is not a replacement of Manual testing.
It enhance and/or supplement manual testing.
Hence it is important that the person writing the automation script should understand the corresponding test case very well.
In my opinion , it is not necessary to have a separate resource for scripting alone.
The output would be good if the person who writes the test case is capable of developing the automation script.

It is better to do both separately.For proper writing of test cases it is better to have good knowledg on that particular domain.After covering all test cases we have to think of what and what not to be automated?Once we have done this we can give to the automation expert.So that he will write good automation code to cover the test cases written .