After an officer involved shooting there are generally two investigations which are conducted.  An internal department or administrative investigation and an investigation conducted by the District Attorney Office or other outside agency to determine any criminal misconduct by the officers.  Though officers cannot refuse to give a statement to their superiors when ordered, they are protected by the Fifth Amendment and cannot be forced to give a voluntary statement for the criminal investigation.
The question officers often face is whether or not they should give a voluntary statement to criminal investigators.  More often than not, these shootings are clean and justified, and officers believe there is no reason not to give a voluntary statement.  Some officers feel that if they don’t give a statement they will be looked at as being uncooperative or as if they are hiding something.  Other officers may believe that if they did something wrong they should be held accountable.   Officers should not be motivated to give a statement for these reasons.
Every shooting is different, and it is not always the case that a voluntary statement should not be given.  However, all the benefits and consequences should be thoroughly weighed before giving a voluntary statement.    Giving a voluntary statement always has the potential to subject an officer to criminal and civil liability, and rarely has any substantial benefits to the officers.   Furthermore, even if the officer gives a voluntary statement, they still maybe ordered to give a subsequent statement to internal affairs investigators.  This is problematic for various reasons, the  most evident is that if these statements are inconsistent, for whatever reasons including a lapse in memory, the officer may subject to allegations of dishonesty and risk their employment.  To avoid this, an officer should always give only one statement and therefore it should be determined if the department is going to conduct an investigation which would require them to give a statement.
Criminal investigators will usually push heavily for an officer to give a voluntary statement immediately after the incident.   This shouldn’t be taken the wrong way, these investigators have a job to do and getting a statement from the officer helps them complete this job.  However, the officer should always protect themselves first by thoroughly reviewing the facts of the case, with a qualified attorney, before giving a statement.   Overall, there should be no rush to give a statement to these investigators, particularly if you are already being ordered to give a statement to your Internal Affairs Investigators.  A copy of the administrative interview can always be given to criminal investigators after it is reviewed by an attorney.   Doing this not only protects the officer, but it has no detrimental consequences to being able to later prosecute the criminal suspects.
In the end the choice is up to the officer as whether or not to give a statement, but this choice should not be made without evaluating all the potential consequences.  Before giving a voluntary statement an officer should always contact their legal representative, and in nearly all cases should decline to give a voluntary statement.