You might find this answer surprising, especially from a former executive at LinkedIn (I'm currently the CEO at Wealthfront).  But the short answer is, yes, in many cases, there is a professional necessity for Facebook.

Let me first get my obvious bias out of the way.  As one of the people who had a hand in building out LinkedIn, I truly believe that there is a valuable and natural separation between your professional identity and your personal identity.  As a result, I would counsel any professional to take great care in how they present themselves on LinkedIn.

However, Facebook is ubiquitous, and it is an easily accessible and common search engine for people.  It is well indexed in search engines like Google.

Chances are, if someone is looking for information about you, as a professional, they will definitely see your LinkedIn profile.  There is also a strong chance they will end up looking for your Facebook profile as well.

Why?

A few reasons really:

• Completeness.  It's easily accessible and fairly common.  It feels like a reasonable part of due diligence.
• Curiosity. Facebook plays strongly to our curiosity to know more about people.  The fact that there is a strong personal / professional split can make people distinctly curious about your split.
• Psychographics.  Whether correct or incorrect, by looking at your social behavior on sites like Facebook, Instagram, Twitter or Pinterest, other professionals may believe they can get a better sense of who you are as a person. 
• Transparency.  Finding a Facebook profile reassures others that you have "nothing to hide."  In a world where Facebook is ubiquitous, not finding a profile or not finding sufficient information may inadvertently beg questions.
  

Facebook offers quite a few tools to control privacy and sharing.  A few recommendations to consider:

• Tightly Control Your Public Profile on Facebook.  To the extent you can, make sure your public profile on Facebook (including public shares and photos) are carefully curated to present the image you would be comfortable with a business partner or colleague seeing.  This doesn't mean hiding your humanity, but it likely means publishing only a limited amount that frames who you are.
• Don't Make Personal Squabbles Public.  Too many times, relatively well-behaved adults present a poor version of themselves when they get into arguments in the comments on Facebook posts, not realizing those comments appear on a public share of a friend.  My advice? From a career perspective, save arguments for private messages and other communication.  Assume that every comment you make on a Facebook share could be public.
  
• Watch for inconsistency.  When people view multiple social profiles, inconsistency (or worse, hypocrisy) tends to stand out.  Presenting yourself as serious and hard-working on LinkedIn, while portraying an active and wild night-life on Facebook will beg questions.

As a final note, Facebook is most useful professionally as an alternate messaging platform for close friends & colleagues.  We live in a world where many people split their attention across too many communication channels.

As a result, I now know people who only reliably respond to text messages, answering in minutes where an email will sit unanswered for days.  I know people who only respond to DMs on Twitter, or messages on WhatsApp.  I also know people who only respond reliably to messages on Facebook. 

So if your goal is fast & efficient communication, it's worth having a channel to your friends & colleagues who prefer Facebook (even if you don't.)

Necessity is a strong word.  You can obviously succeed professionally without Facebook.  However, I would argue that using Facebook effectively has distinct value as a professional and there are situations where it can be considered a necessity.