My school district is suggesting a Facilitator. How is the facilitator chosen? Would I have a say in this? 

I'm in Ohio.

What's the difference between facilitation and mediation?

I'm not sure how Ohio does this but in PA, they have a pool of facilitators and the state assigns them.  I don't believe either the school of the family have a say.

Ohio does have an info sheet on facilitators:  https://education.ohio.gov/getattachment/Topics/Special-Education/Dispute-Resolution/Facilitation_Handout.pdf.aspx

They also have a sheet on mediation and it does have a lot of parallels:  https://education.ohio.gov/getattachment/Topics/Special-Education/Dispute-Resolution/ODE_MediationHandout-8-17-18-1.pdf.aspx?lang=en-US

It does look like schools/families do pick who is used to facilitate/mediate in Ohio.  I believe that mediators are looking for an agreement on what should be in the IEP where a facilitator keeps the meeting on track but doesn't push them to come to an agreement.

Does the student attend both of these types of meetings?

An older student (age 14+ in PA; 16+ per IDEA) should be invited to these meetings.  If they go depends on a number of things.  My one child could not deal with anything that's negative so she stayed for a bit & left before anything negative was said.  Mediation can take days and it's not appropriate for a student to miss that much school IMO.

In Ohio, the school or parents can request mediation or facilitation (by calling 877-644-6338 or emailing [email protected]), but both parties have to agree to participate in it. Usually ODE will send bios of around 3 mediators/facilitators to choose from. You and the district would need to agree on the mediator/facilitator. Mediation/facilitation are similar. If an agreement is reached in mediation, the mediator can write up a legally binding agreement for both parties to sign. A facilitated IEP meeting can be helpful in contentious relationships as the facilitator should ensure all voices are heard and that the meeting stays on schedule. Some facilitators are more fair to families than others (giving them time to share their concerns/ideas), but they're all technically supposed to be neutral third parties. School staff may be more polite to you if an ODE facilitator attends your meeting. Mediation sessions may run much longer than facilitated IEP meetings.

I'm also wondering how to prepare for a facilitated meeting.  Should I have a statement prepared?  A list of questions that are continually unanswered?  

@Laura- Have you already submitted your letter of parent concerns to the district? If so, and if you have the contact info for the facilitator, share a copy of that letter with the facilitator as well. 

What happens after the facilitated meeting? 

Does the facilitator write or share a report with anyone? Or, is the information just filed away?

I am a facilitator in Missouri, so can only speak to how facilitated IEP's are conducted in my state, but I agreed with Jenna that all you need to prepare is a letter (or email) of parent concerns.  In Missouri, I speak with both the parents and the school district via phone prior to the meeting.  In my phone conversation with the parents I try to pull out all concerns that they have.  I also offer to draft the parent concerns letter is they are unsure of how to do this.  I, of course, have them review and approve the list.  After parent approval, I forward a copy to the school district.  It makes the process go a little smoother when the district is aware of the parent concerns prior to the meeting, as is the case with any IEP meeting.

The meeting itself will most likely be conducted similarly to a "regular" IEP meeting.  In fact, the facilitator has to follow the agenda outlined in the IDEA.  There will just be some extra "bells and whistles" in the meeting such as visual charts that the facilitator (or someone he/she asks) tracks.  The neutral facilitator will only jump in if tensions are rising, the meeting is getting offtrack, pros and cons need to be developed to reach a consensus on a particular proposal, etc.

As for a report and what is done with the information at the end of the meeting, in Missouri there is no report generated after a facilitation.  The school district is allowed to take pictures of the charts if they do so with a school device (not their personal cell phone) for purposes of adding it to the student's file.  They charts are then taken down and destroyed.